Form and Process in Music, 1300–2014: An Analytic Sampler [1 ed.] 1443885509, 9781443885508

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Form and Process in Music, 1300-2014

Form and Process in Music, 1300-2014: An Analytic Sampler Edited by

Jack Boss, Heather Holmquest, Russell Knight, Inés Thiebaut and Brent Yorgason

Form and Process in Music, 1300-2014: An Analytic Sampler Edited by Jack Boss, Heather Holmquest, Russell Knight, Inés Thiebaut and Brent Yorgason This book first published 2016 Cambridge Scholars Publishing Lady Stephenson Library, Newcastle upon Tyne, NE6 2PA, UK British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Copyright © 2016 by Jack Boss, Heather Holmquest, Russell Knight, Inés Thiebaut and Brent Yorgason and contributors All rights for this book reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. ISBN (10): 1-4438-8550-9 ISBN (13): 978-1-4438-8550-8

TABLE OF CONTENTS

Preface ........................................................................................................ ix Jack Boss PART I: MUSIC OF THE FOURTEENTH, EIGHTEENTH AND NINETEENTH CENTURIES Chapter One ................................................................................................. 3 Metric Dissonance and Greater Metric Dissonance in Late FourteenthCentury Music Timothy Chenette Chapter Two .............................................................................................. 21 Structural Cyclicity in Trecento Ballate Heather Holmquest Chapter Three ............................................................................................ 39 Mozart’s Common (yet Uncommon) Common-Tone Transfers Susan K. de Ghizé Chapter Four .............................................................................................. 53 Analyzing Wagner’s “Der Engel”: Questions Posed after Application of Recent Transformational Theories Barbora Gregusova Chapter Five .............................................................................................. 81 Maus and the Meter Cycle: Three Narrative Analyses Brent Yorgason PART II: MUSIC OF THE TWENTIETH CENTURY Chapter Six ................................................................................................ 97 Voice Leading and Musical Spaces in Britten’s Opus 70 Dale T. Tovar

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Chapter Seven.......................................................................................... 111 Ending Ligeti’s Piano Etudes Sara Bakker Chapter Eight ........................................................................................... 125 In Disguise: Borrowings in Elliott Carter’s Early String Quartets Laura Emmery Chapter Nine............................................................................................ 147 Compositional Spaces in Mario Davidovsky’s Quartettos Inés Thiebaut Chapter Ten ............................................................................................. 175 Formal, Impulse, and Network Structures in Donald Martino’s Impromptu No. 6 Aaron J. Kirschner Chapter Eleven ........................................................................................ 197 Exploring New Paths Through the Matrix in Ursula Mamlok’s Five Intermezzi for Guitar Solo Adam Shanley PART III: POP MUSIC, JAZZ, AND ANCIENT AND SPECULATIVE MUSIC THEORIES Chapter Twelve ....................................................................................... 227 “Little High, Little Low”: Hidden Repetition, Long-Range Contour, and Classical Form in Queen’s Bohemian Rhapsody Jack Boss Chapter Thirteen ...................................................................................... 255 Schenkerian versus Salzerian Analysis of Jazz Rich Pellegrin Chapter Fourteen ..................................................................................... 275 A Critical Comparison of Aristoxenus’ and Ptolemy’s Genera Matthew E. Ferrandino

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Chapter Fifteen ........................................................................................ 289 Spaciousness or Evenness? A Theory of Harmonic Density in Analyzing 20th- and 21st-Century Music Yi-cheng Daniel Wu Contributors ............................................................................................. 323

PREFACE

During the past seven years, the West Coast Conference of Music Theory and Analysis has published three volumes of conference papers with Cambridge Scholars Publishing. In 2008, Musical Currents from the Left Coast was released, based on the papers given at our meeting at the University of Utah in 2007. Our first book surveyed and analyzed music in a variety of styles and using numerous approaches, with a closing symposium that explored Schoenberg’s Op. 11 Piano Pieces from four substantially different perspectives. It has influenced research in music theory significantly, continuing to receive mention in articles and conference presentations to this day. In January of 2013, we published Analyzing the Music of Living Composers (and Others), based on presentations from our meeting at the University of Oregon in 2010. In our second book, we focused on “applying traditional music-analytic techniques, as well as new, innovative techniques, to describing the music of composers of the late 20th and early 21st centuries.” The book also included analyses of music of earlier eras that we saw as influential for contemporary composers. In time, we believe Analyzing the Music of Living Composers will have an impact on music scholarship even stronger than its predecessor, and could also influence the art of music composition in important ways. Now we are pleased to offer this third book in the series, which is drawn from papers presented at our 2014 conference, again held at the University of Utah. Proposals for the 2014 meeting spanned a wider spectrum of musical styles than we had ever seen before. We had originally called for papers on European twelve-tone music after the Second World War, but we were also able to schedule sessions on fourteenth-century music, pop music and jazz, the music of living composers, narrative and characterization, and the history of music theory. The title of our book reflects the large span of musical cultures and styles that are represented within, but also accounts for the common thread through all of these essays, a strong emphasis on understanding the forms and processes of the music through analysis. The book divides into three main sections, which correspond to the roughly equal divide during our conference between music from prior to the 19th century and music of the 20th century, with a handful of papers on popular music, jazz, the history of music theory and speculative music theory.

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In Part I, we begin with two chapters that explore 14th-century music from different perspectives. Timothy Chenette discusses heard meters in polyphonic music and the kinds of “metric displacements” that emanate from them, while Heather Holmquest focuses her attention on monophonic songs from the Rossi and Squarcialupi codices, and the cyclical melodic patterns that can be heard within them when one adopts a modified Schenkerian approach. Susan de Ghizé then carries us forward to the 18th century with her study of the diverse range of common-tone transfers in the Mozart Piano Sonatas. Finally, the last two chapters of part I approach 19th-century music from contrasting viewpoints. Barbora Gregusova studies structural cohesion and text painting in Wagner’s “Der Engel” using the tools of transformational theory, while Brent Yorgason performs narrative analyses (following Fred Maus) of the music of Brahms, Tchaikovsky, and Schumann, using “meter and expressive timing as the basis of the plot.” Contemporary music has traditionally been a favorite topic at West Coast Conference meetings, and the 2014 meeting was no exception. Several scholars responded to our call for papers on postwar European music, but a variety of late 20th-century (and some slightly older) musical styles were represented as well. Part II of our book begins with a chapter by Dale Tovar on the use of octatonic collections and ordered pitch-class interval cycles in Benjamin Britten’s Nocturnal after John Dowland. This is followed by five chapters on more recent composers, reminding the reader of our second book Analyzing the Music of Living Composers (and Others). Sara Bakker considers simultaneous offsetting rhythmic ostinati in Ligeti’s Piano Etudes that create “cycles” too long for the duration of the pieces, and shows how Ligeti makes adjustments to the ostinati to create convincing cadences. Laura Emmery explores Elliott Carter’s string quartet sketches, showing how they demonstrate processes of borrowing from composers such as Bartók and Webern. Inés Thiebaut and Aaron Kirschner describe the various patterns and processes that constitute the aggregate and serial organizations of Mario Davidovsky’s Quartetto (Thiebaut) and Donald Martino’s Impromptu No. 6 (Kirschner). To bring Part II to a close, Adam Shanley explains how Ursula Mamlok uses the twelve-tone matrix in unique and creative ways to create the pitch organization for her Five Intermezzi for Guitar. Part III collects together four chapters that represent the variety of other topics that were discussed at our meeting. I begin in the realm of analyzing popular music, with my account of Freddie Mercury’s Bohemian Rhapsody. I utilize Schenkerian and Neo-Riemannian analytic techniques, as well as allusions to traditional sonata form, to show how the song is a surprisingly unified structure, rather than a rhapsody, and how its large contour expresses its underlying meaning. Rich Pellegrin looks at ways in which the analysis of

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jazz can be aided by using a “Salzerian” approach that relaxes restrictions on what may be admitted as a structural harmony or line, as opposed to a strict Schenkerian approach. Matthew Ferrandino turns our attention toward ancient music theory, with his consideration of the “dichotomy between theory and practice” represented by Aristoxenus’s Elementa harmonica and Ptolemy’s Harmonics. Finally, Yi-Cheng Daniel Wu brings the book to an end with a speculative theoretical chapter that reconsiders interval-class space as well as criteria for “evenness” and “spaciousness,” and then develops a measurement “testing the degree of chromaticness of a chord.” Our third book should be of great interest to many of the same groups that were targeted by our first two volumes. First, musicologists and music theorists who work on any of the diverse styles of music that we explore. Then, performers of these various kinds of music (numerous chapters include advice for interpreters that grows out of the analytic findings). Finally, our audience will hopefully include music lovers who are seeking ways to enhance their listening experience by understanding more of the musical forms and processes that organize their favorite pieces. Jack Boss, December 2015

PART I: MUSIC OF THE FOURTEENTH, EIGHTEENTH AND NINETEENTH CENTURIES

CHAPTER ONE METRICAL CONSONANCE, METRICAL DISSONANCE, AND GREATER METRICAL DISSONANCE IN THE ARS SUBTILIOR TIMOTHY CHENETTE

The notation of meter, or mensuration, in the c. 1400 Ars subtilior uses multiple levels, of which the two most salient are tempus and prolation. Just as modern time signatures designate whether the number of beats in a measure is two, three, or four, tempus designates whether the number of semibreves in a breve is two or three; and just as modern time signatures indicate whether the subdivision of the beat is duple (simple meter) or triple (compound meter), prolation indicates the number of minimae in a semibreve—again, two or three. Editors of modern editions typically use these analogies to determine time signatures. For example, perfect tempus (three semibreves in a breve) with minor prolation (two minimae in a semibreve) is typically transcribed in #4(three beats in a measure, two subdivisions in a beat). Thus, inasmuch as notation constrains or suggests what metric and rhythmic experiences are possible, it might seem that the two systems would have significant similarities. Yet the earlier system has flexibilities that are not inherent to modern practices. In common-practice music, the tyranny of time signatures constrains composers’ ability to make alterations to meter at the level of the beat or the measure, though changing groupings are common at the levels of subdivision (through tuplets) and of hypermeter (through varying phrase lengths). In fourteenth-century music, which does not make use of barlines and measures, composers are constrained by the levels that I described above, but these differ from time signatures in several ways, the most prominent of which are listed below.1 • First, notes may be colored red, indicating that they lose one third of their duration: this is called coloration. This gives an effect similar to hemiola: in the classic instance, black semibreves in

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imperfect tempus with major prolation (the rough equivalent of dotted quarter notes in ^8) are transformed into red semibreves, suggesting perfect tempus with minor prolation (the rough equivalent of quarter notes in #4). • Second, a metric unit may have another complete metric unit (or more than one) inserted within it: this is called syncopation. • Finally, different voices of a polyphonic piece may be written in different mensurations: this is sometimes called polymensuralism. In part because of these devices of coloration, syncopation, and polymensuralism, scholars have usually emphasized the distance between modern notions of meter and those of the fourteenth century. For example, Jason Stoessel, who made the excellent editions used in this essay, prefaces them with the justification that “[the tick bar line’s] advantage over the internal bar line on each staff lies in its minimal implication of a regularity that is central to the concept of the bar line in today’s common practice notation” (2002, 11).2 Catherine Hawkes’s doctoral thesis is on syncopation: she points out that, unlike modern syncopation or accents considered against an underlying meter, fourteenth-century metric groups retain their integrity in notation pedagogy even when displaced through syncopation. This, she says, “leads to the conclusion that the intervening notes need not be performed any differently than they would be performed if the syncopation were not there” (2009, 160). Uri Smilansky’s dissertation at one point focuses on a passage written in one mensuration but seeming to imply another through repetition; modern meter pedagogy would suggest a different accentuation in a different meter, but Smilansky concludes that it is impossible to know if this would have been true in the fourteenth century (2010, 166). While it is valuable to come at this repertoire without anachronistic preconceptions, modern cognitive theories of meter suggest that part of any musical experience—regardless of cultural-historical background—is an automatic attempt to entrain ourselves to a single sense of meter. Justin London, in his book Hearing in Time, argues that meter is “not fundamentally musical in its origin,” since it relies on the human capacity for entrainment, “the synchronization of our attention with our capacity and preparedness for movement,” and thus is potentially universal (2012, 4 and 12). For this reason, London argues that across cultures, meters are “subject to the same basic formal and cognitive constraints” (2012, 7). Yet “synchronizing our attention with our capacity and preparedness for movement” is more abstract than most modern definitions of meter, which tend to emphasize either spatial metaphors and hierarchy within the

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music or a sense of recurring “accent.” For this reason, when I discuss meters in this music, I do not mean to assert that listeners of the time would have conducted, counted, or otherwise moved as we do. Rather, I assert that there is something in this music that facilitates prediction of important events such as the arrival of consonances and synchronization of attack points at regular intervals. Whether we try to imagine ourselves as early musicians or simply use our more familiar listening strategies, we will be using this innately human capacity for prediction and accessing something that truly is in the music and seems designed to stimulate this capacity. I will begin by revisiting coloration, syncopation, and polymensuralism, and I will suggest that a useful way to conceptualize these manipulations is through the concept of “metric dissonance” as theorized by Harald Krebs. Then, in the second part of the chapter, I will analyze two pieces in detail: these pieces will show that repeating patterns do indeed sometimes invite entrainment to a single underlying meter; but that in certain cases, unique aspects of fourteenth-century notation allow for aesthetic trajectories that we may miss if we are not attuned to the unique aspects of mensural notation. Particularly, dotted-quarter-note and quarternote beats are often simultaneously available to the listener, or may alternate, in ways that create much of the beauty in this repertoire.

Coloration, Syncopation, and Polymensuralism as Metric Dissonance In his book Fantasy Pieces: Metrical Dissonance in the Music of Robert Schumann, Harald Krebs notes the nineteenth-century “conceptualization of musical meter as a set of interacting layers of motion, each layer consisting of a series of approximately equally spaced pulses” (1999, 22). This description has obvious connections to modern notions of meter, which define subdivisions, beats, and measures, as well as potentially other levels; it also has precedents in Medieval metric practice, with the interacting layers of prolation and tempus (and modus, though this level is often not emphasized in the music). Given this essay’s focus on perception, and the skepticism in the scholarly community surrounding modern assumptions about pulse in early music, it will be useful to reframe this: instead of layers of pulses, we might imagine different rates of motion, each created by recurrence of important events that facilitate entrainment at different levels. Krebs’s concepts of “metrical consonance” and “metrical dissonance,” similarly reframed, are also useful here: states of metrical consonance are common enough in this repertoire that certain

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aspects of modern listening may be brought to bear, while Krebs’s two types of metrical dissonance map nicely onto coloration and syncopation. I will show these connections and examine the role of polymensuralism in complicating the idea of a “primary metrical consonance” to set the groundwork for the analyses that follow. Metrical consonance exists when interpretive layers are in alignment, that is, a pulse (or, reframed, an attentional peak) at any given level coincides with a pulse at each faster level. Pieces that emphasize a state of metrical consonance are common in the Ars subtilior. The opening of Conradus de Pistoria’s Se doulz espour (Example 1-1) is such an example: if the listener uses the first two attacks (m. 1 and m. 2) to project a third important event the same amount of time later, they will be rewarded by simultaneous attacks in the lower parts (m. 3), then the arrival of these parts on a perfect consonance (m. 4), etc., suggesting a continuous layer of motion. This layer aligns with faster layers: the attacks on every transcribed quarter note except m. 1, beat 2, and the near-constant running eighth note composite rhythm. The meter is secure: the use of transcribed eighth notes, minims, is limited to the initiation and conclusion of smallscale syncopation or to fill exactly half a transcribed measure.

Example 1-1, Conradus de Pistoria, Se doulz espour, opening. Score from Stoessel 2002, 276.

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This piece is, of course, not without metric interest, as annotated in the example. The cadence to A and E at the downbeat of m. 4 is prepared with the tension of syncopation in the upper two voices; similarly, the cadence to G in m. 6 is prepared by a longer top-line syncopation. The annotations indicate, in labels derived from Krebs, that a metric layer that is the same length as a layer of the “primary consonance”—in this case, one quarternote long—has been briefly displaced against that consonance. The “D” stands for “displacement dissonance,” one of Krebs’s primary types of metric dissonance; “2+1” indicates that a cycle of 2 units has been displaced by one unit; and “1 = e” designates that the unit used to measure these durations is the eighth note. Still, these disruptions constitute simply minor metric dissonance against a clearly defined primary meter: they always “resolve” quickly at the cadence, and the tenor consistently articulates the “aligned” quarter-note beat. It is worth noting how well this type of Medieval syncopation maps onto Krebs’s displacement dissonance. Johannes de Muris defines syncopation (sincopatio) as “a thorough division of a figure through separate parts which are reduced one to another by numbering perfections.”3 A “figure,” in this case, is a duration that would fill a metric unit at some level; in syncopation, it is “divided,” and “separate parts,” or other complete metric units, are inserted. In the top line of mm. 4–5 in Example 1-1, for instance, the figure in question is the length of a quarter note; it has been divided into an eighth rest and an eighth note, between which have been inserted three whole quarter notes. In theory, this need not be the case, but in practice, as here, the inserted units nearly always clearly articulate a layer of motion that is therefore displaced, and syncopation in the Medieval sense that is longer than a single inserted note can virtually always be described coherently with Krebs’s displacement labels. The texture becomes much more confusing in the second line. We get the longest phrase yet, nearly matching the length of the previous two combined before cadencing for the first time to the piece’s final, D, in m. 11 (not shown); in the upper voices, syncopation is no longer limited to just precadential decoration; and the first tenor syncopation serves to undercut the metric foundation. This is about as confusing as this piece’s meter gets, but the continuity of the primary consonance—roughly, a modern @4—before and after this phrase render it, again, metric dissonance against a clear primary meter. This opening as a whole follows an interesting metric path, gradually becoming more and more metrically dissonant before each cadence, but ultimately the piece presents no

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significant problems to a modern notion of meter, and it would be rather easy to conduct in duple meter throughout (if one wished). While displacement dissonance maps nicely onto syncopation, Krebs’s “grouping dissonance” maps onto coloration. Example 1-2 gives the cadences of the major sections of Philippus da Caserta’s En atendant soufrir. In each case, there is a clear layer of motion articulating the transcribed dotted quarter note, or black semibreve, and there is a simultaneous layer of transcribed quarter notes, or red semibreves. (The red color is indicated in this and most modern editions by open brackets above.) These layers are not the same length, but rather group the underlying eighth-note (minim) pulse differently, and thus Krebs would consider them together a “grouping dissonance”: G2/3 (1 = e), indicating a conflict between one layer that groups the minims into twos and one that groups them into threes.

Example 1-2a, Grouping dissonances at major cadences in Philippus da Caserta, En atendant soufrir. From Stoessel 2002, 76–77. End of section A.

Example 1-2b, Grouping dissonances at major cadences in Philippus da Caserta, En atendant soufrir. From Stoessel 2002, 76–77. End of section B.

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Example 1-2c, Grouping dissonances at major cadences in Philippus da Caserta, En atendant soufrir. From Stoessel 2002, 76–77. End of section C.

In one way, polymensuralism also represents grouping dissonance. The three different simultaneous mensuration symbols at the beginning of Antonello da Caserta’s rondeau Dame d’onour, c’on ne puet esprixier, shown in Example 1-3, represent grouping dissonances: the top two parts would be labeled G3/2 (1 = q.), and the outer two G3/2 (1 = e). Yet here

Example 1-3, Polymensuralism, but at a low level of metric dissonance, in Antonello da Caserta, Dame d’onour, c’on ne puet esprixier. Based on Le composizioni 2005, 137.

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these grouping dissonances are more apparent in the notation than in the sound, as the cantus’s transcribed ^8is clearly primary. The tenor’s #4 has the same periodicity—that is, its downbeats align with those of the cantus—and moves so slowly that among the first twelve measures of transcribed music, only three even mildly contradict the compound duple division of the measure, the last of which is a typical pre-cadential hemiola. The cantus, in contrast, clearly articulates its designated meter throughout, never engaging in even a single syncopation. Finally, the contratenor clearly articulates a dotted-quarter-note beat, as one might expect, and does not clearly privilege measures of 98over groupings that follow the cantus’s ^8. The level of metric dissonance is very low throughout. Though polymensuralism is not extremely common, and though it can be used in a way that keeps the level of metrical dissonance low, its very possibility also represents a challenge to the concept of the “primary metrical consonance.” In Krebs’s formulation, “One of the metrical interpretive layers generally assumes particular significance for the listener. . . . The layer formed by these pulses frequently, though not always, occupies a privileged position in the score, being rendered visually apparent by notational features such as bar lines and beams” (30). Thus the terms consonance and dissonance, to Krebs, do not just refer to the literal “sounding together” of aligned layers and “sounding against” of nonaligned layers, but also suggest an analogy to pitches in counterpoint, where more dissonant states are expected to “resolve” to more consonant states, and in the end, to the “primary metrical consonance”—analogous to the tonic. And usually, this primary metrical consonance is indicated by the meter signature, which in tonal music must be the same in all simultaneous parts. The possibility of polymensuralism, then, provides a conceptual model where the default state of a piece of music may not, ultimately, be a metrical consonance, but a dissonance. (As will be seen below, this state often arises even when simultaneous parts are written in the same mensuration.) When this is the case, and when it is reinforced perceptually, we might call this state the “primary metrical dissonance.” Analogies to resolution may still be made, however, as this primary metrical dissonance may be enriched and complicated by more complex dissonances, that in the end resolve (ironically) to this lesser degree of dissonance. One final theoretical point remains, given the perceptual focus of this essay, which is to reconcile London’s statement (reporting on perception and cognition research) that “there is no such thing as a polymeter” (67) with this emphasis on polymensuralism as a possible default state for a

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piece of music. London’s pronouncement indicates not that there cannot be multiple simultaneous meters written in the music, but that humans cannot simultaneously synchronize their attention to multiple conflicting cycles: rather, we will focus on one, measuring the other against it. From a perceptual standpoint, then, I do not mean that our attention is evenly divided, since this research suggests it cannot be. Instead, I mean that evidence in the music may draw us in two different ways, and that a kind of cognitive dissonance arises as fluctuations in the written music or performance factors draw our attention now to one cycle, now to the other, making us vacillate between. As will be seen below, this vacillation is exploited in the metric progressions of certain pieces.4

Analyses The two pieces analyzed above, Se doulz espour in @4and Dame d’onour in ^8, each use a single, clear meter (perceptually, at least) that is recognizable to modern musicians and listeners. This is an important point: aspects of modern metric practice may be brought to bear on this repertoire without anachronism. In addition, this creates an environment where frustration of these metric expectations will be more effective. I will spend the rest of this essay looking at two pieces that are far more complicated and more fully take advantage of the kinds of flexibility offered by the mensural system to create unique metric progressions. The basic contrast of metric strands in Philippus de Caserta’s ballade En atendant soufrir is encapsulated in its section-ending cadences, shown in Example 1-2 above. In each, at least one voice articulates dotted quarter notes, and at least one articulates quarter notes. (In the B section, there is also a syncopation in the top voice.) These are not mere precadential hemiolas: transcribed quarters and dotted quarters conflict throughout the piece, generating a “primary metrical dissonance.”5 Because these points of repose do not “resolve” to one of these strands, I will treat both as “underlying continuities”—as potential meters—with performance choices and slight differences of texture determining which one listeners attend to as primary at any given time. This clearly differs from modern metric practices, where a written time signature would generally dictate the “true” meter: using fourteenth-century notation, Philippus does not need to decide, and can leave it up to the performers and vagaries of performance circumstances. The metric conflicts in En atendant seem designed to bring attention to small-scale contrasts. Much of the opening of the piece (Example 1-4) jumps quickly back and forth between these two apparent beats. Mm. 4–5

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clearly articulate both layers, then mm. 6–7 clearly express only dotted quarter beats. Starting in m. 8, the

Example 1-4, Quarters (imperfect semibreves) vs. dotted quarters (perfect semibreves) in the opening of Philippus da Caserta, En atendant soufrir. From Stoessel 2002, 74.

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quarter-note layer is strongly articulated by a repeated pattern in the cantus, which is gradually liquidated in mm. 10–11, losing its characteristic eighth notes. Finally, no attack whatsoever materializes on the downbeat of m. 12: this is particularly surprising because the hemiola of the previous measures, and the E-G-Bb sonority right before, strongly hint at a D-A cadence here. (The cadence finally arrives a measure later.) This small-scale contrast, these quick changes of metric emphasis, keep us from designating one meter as primary: though the prominent tenor is always steadily singing dotted quarter notes, the cantus’s equally prominent and florid melody is largely based on quarter notes after m. 7, and aspects of performance and a listener’s predisposition will likely influence which is more clearly attended to. The opening of the piece gives a clue to yet another level of metric conflict: repeated patterns here seem to articulate measures of 98, rather than the transcribed ^8. Indeed, just as quarter notes and dotted quarters compete at the level of the beat, there is a constant alternation of apparent ^8measures and 98measures at the level of the measure. After two apparent 98measures, the periodicity of the quarter/dotted-quarter conflict in transcribed m. 4 seems to confirm ^8/ #4; mm. 5–8 again suggest 98, with only the inner-voice contratenor articulating the downbeat of m. 6; and then ^8/ #4 is very clearly confirmed by the repeated patterns in mm. 8–11. Though not shown in the example, a long passage of clear ^8/ #4 follows, and then yet another passage of 98in mm. 19–21 accompanying the words “et en langour,” or “and in languor.” The interaction of these two levels of conflict, quarter vs. dotted quarter beats and measures of 98vs. ^8/ #4, is particularly fascinating. When the apparent measure is six eighth notes long, the conflict of quarters and dotted quarters is nearly always foregrounded as the periodicity of this conflict confirms this length of measure. When the quarter notes recede or disappear, the more languorous dotted quarters nearly always stretch the measure to its longer state. These two levels of metric conflict and their interaction create constant fluctuations in our sense of meter. Aesthetically, this results in a feeling of yearning and affected artifice, as individual metric states appear as brief illusions but never become satisfactorily established. In as much as the extremely high, constant level of metric dissonance is ever resolved, this happens at the cadences; but, again, the preparation for each cadence clearly states the quarter/dotted quarter conflict in a very strict manner. Along with the subjugation of the normally quite florid upper line into a simple articulation of this conflict, this “resolution” of metric conflict feels to me like resignation to the fact that it will never be resolved.

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The opening of my next example, Bartholomeus de Bononia’s Arte psalentes, is particularly complex and ambiguous. In Example 1-5, rounded rectangles mark passages that articulate the dotted quarter note layer, and square rectangles mark passages that articulate the quarter note layer. The opening measure in cantus and contratenor clearly divides the transcribed “measure” in half, and the longest stretch of consistent articulation in the cantus also indicates ^8. The cantus’s metrical layer, however, is displaced against the ^8barline (as measured by tick bar lines in the lower parts), and it is overlaid not only against occasional quarter notes in the contratenor and perceptually salient tenor but also much more complicated rhythms in the contratenor. In fact, in the first 16 measures of the piece, about the only consistency that can be found is a prevalence of quarter notes right before cadences. (Only one of these is shown in the example, at the beginning of the second line: the extreme length of phrases is another contributor to the metric complexity.)

Example 1-5, Complex opening of Bartholomeus de Bononia’s Arte psalentes.6 Rounded rectangles indicate passages that articulate the dotted-quarter layer; square rectangles indicate the quarter layer. From Stoessel 2002, 110.

This makes it all the more surprising when the lower parts in mm. 17–20 (Example 1-6) suddenly come together and coordinate with the upper part in a kind of call and response as the cantus engages in a literal sequence, all

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clearly indicating dotted quarter note beats. The remarkable nature of this passage is highlighted by changes in the tenor: thus far it has nearly always moved within the fifth between G3 and D4; in m. 16, a dramatic rising line brings it all the way up to A4 for the beginning of this passage.

Example 1-6, Dotted-quarter layer in Arte psalentes, mm. 17–20. From Stoessel 2002,111.

Example 1-7, Quarter-note layer in Arte psalentes, mm. 21–24. From Stoessel 2002, 111.

Immediately following, in mm. 21–24 (Example 1-7), the texture switches to only clear quarter-note articulations, and again, there is a sequence in the cantus. The parts trade off eighth notes to create a continuous texture and to emphasize, in the moments of swapping those eighth notes between parts, the quarter-note beats.7 This passage is the last portion of the “body” of section A: after the cadence in mm. 25 and 26, immediately following the passage in the example, there is a 9-measure closing that is repeated literally at the end of the piece, as is traditional for a ballade. In this closing, one more brief passage of consistency appears, as mm. 29–31 clearly articulate only dotted quarters. Aside from the pleasure of following these changes, part of the reason for them may be to emphasize important portions of the text. The text translates as, “Let us praise with art the goodness of the Fathers in the

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presence of the sovereign pontiff, with serene countenance, may the dignity of the master deign to guide the singing of the young pupil.” The dotted-quarter passage begins at the word for “of the Fathers,” and the passages together lead up to a cadence on the word “Pontiff.” The text of the next passage continues, “and if the young pupil’s singing be lacking in skill”; perhaps to suggest or even challenge a lack of skill, this section nearly always juxtaposes clear dotted quarter and quarter beats. Mm. 44–47 are a good example: here the cantus uses eighth notes only to fill quarter-note beats except for a brief syncopation, while, in the tenor, repeated groupings of eighth notes and quarter notes clearly suggest dotted-quarter beats. This continues for three measures past the section I have excerpted here in Example 1-8.

Example 1-8. Metric dissonance in the B section of Arte psalentes. From Stoessel 2002, 112.

The final section completes this thought: “may it please you to teach him the art of true song,” and finally we have a section that is relatively stable in its use of dotted-quarter-note beats. After an extremely confusing passage in mm. 61–64, full of displaced seemingly metric strands, the rest of the piece clearly emphasizes dotted-quarter beats except in cadential preparations. This emphasis on dotted quarters continues in the last nine measures of the piece, which, again, are directly repeated from section A. The piece as a whole thus moves from a state of extreme confusion (hardly even a stable “primary metrical dissonance”), through brief states of extreme metrical consonance, to a moderate level of metrical consonance at the end.

Conclusion I will briefly review some premises in light of these two analyses and the preceding theoretical discussion. I began by arguing that meter is a part of the listening experience of any music, and repeated patterns in this rep-

Ars subtilior

17

ertoire do indeed invite us to entrain to a recurring cycle of beat and, often, measure. Second, notation both constrains and suggests the kinds of metric experiences that are possible. Ars subtilior mensural notation invites composers to employ metric dissonance through syncopation (displacement dissonance), coloration (grouping dissonance), and polymensuralism. In addition, the model of polymensuralism suggests the possibility of pieces where the ultimate state of the music is actually metric dissonance, not metric consonance, providing an ideal context for the yearning and affected artifice of courtly love poetry. I will close now with a few ramifications for performance and listening. First, in certain ways our modern notions of meter are not so far from those of the late fourteenth century, especially in pieces with regular articulation of one specific beat type; for this reason, it may not be totally anachronistic when ensembles trying desperately to perform this music accurately choose a single meter to tap or conduct. Second, while some flexibility in performance is nearly always desirable, it will be helpful to keep in mind the distinction between quarter and dotted quarter beats aligned with transcribed measures, which may together generate a “primary metrical dissonance,” and displaced beats or beats of other lengths, which tend to be of short duration and may perhaps be performed with more freedom without destroying the sense of the piece. Finally, in contrast to modern time signatures that tend to decree a single sense of meter that is operative throughout an ensemble, a performance of or listening to this repertoire will be the richer for paying close attention to the conflicts and alternations between quarter notes and dotted quarter notes: as the analyses I described suggest, these conflicts often create interesting trajectories and affective states. In a repertoire so obsessed with complex rhythm, this allows us to accept the confusion these create, and to embrace it as beautiful.

Notes 1

Instead of being given visual groupings through measures and beamed-together beat units, performers were expected to learn the process of “reduction” (reductio)—that is, scanning the music to count note values and group them together appropriately at each level. This grouping process is very important, because in certain cases, primarily in “imperfection” and “alteration,” the matter of which notes group together can affect their duration. 2 The editions comprise Volume 2 of Stoessel’s dissertation. Those wishing to understand the examples in this essay in context can download this volume for free at http://diamm2.cch.kcl.ac.uk/resources/stoesseldiss.html. The only example in this essay not based on Stoessel’s scores is Example 1-3, prepared by myself.

18

3

Chapter One

Sincopa est divisio cujuscunque figure per partes separatas que numerando perfectiones ad invicem reducuntur. (This text is adapted from a number of sources that can be found at the Thesaurus Musicarum Latinarum: see http://www.chmtl.indiana.edu/ tml/14th/14TH_INDEX.html.) 4 Chapter 4 of Krebs’s book, “Metrical Progressions and Processes,” uses these terms somewhat technically and defines large-scale progressions of metric states analogous to large-scale pitch processes in tonal music. My use of these terms is meant to evoke his, in the sense that we are both arguing for the importance of rhythmic/metric trajectories in the artistic appreciation of our respective repertoires, but the progressions described here are smaller in scale. 5 This despite the fact that the parts are not written in different mensurations. In fact, there are no mensuration signs given at the beginning of the music (this is common at the time), though it is clear from context that the interpretation of each part relies on the assumption of imperfect tempus, major prolation (roughly, ^8). 6 The red notes in mm. 2–3 of the superius are odd, in that they are applied to normally imperfect rather than perfect semibreves. In his compendious manual on interpreting early notation, Willi Apel explains this usage: “Although coloration usually diminishes the value of a note (by one third), it is occasionally used in an opposite meaning, signifying an increase by one half, that is, synonymous with a dotted note. Naturally, this type of coloration can only be applied to imperfect notes” (406). 7 Part of the beauty of this passage lies in a motivic foreshadowing in m. 10, which states the model of this sequence but does not take it further.

Ars subtilior

19

Works Cited Apel, Willi. 1949. The Notation of Polyphonic Music 900–1600. 4th ed. Cambridge, MA: The Mediaeval Academy of America. Le composizioni Francesi di Filippotto e Antonello da Caserta tràdite nel codice Estense Į.M.5.24. 2005. Edited with commentary by Carla Vivarelli. Diverse voci, no. 6. Pisa: Edizioni ETS. Hawkes, Catherine. 2009. “Syncopation in the Fourteenth and Fifteenth Centuries: A Review of Treatments of Syncopation in French and Italian Treatises and a Study of Contemporary Musical Examples that Display the Use of Syncopation in Various Contexts.” D.M. thesis, Indiana University. Krebs, Harald. 1999. Fantasy Pieces: Metrical Dissonance in the Music of Robert Schumann. New York: Oxford. London, Justin. Hearing in Time. 2012. 2nd ed. New York: Oxford. Smilansky, Uri. 2010. Rethinking Ars Subtilior: Context, Language, Study and Performance. Ph.D. diss., University of Exeter. Stoessel, Jason. 2002. “The Captive Scribe: The Context and Culture of Scribal and Notational Process in the Music of the Ars subtilior.” 2 vols. Ph.D. diss., University of New England, Australia.

CHAPTER TWO STRUCTURAL CYCLICITY IN TRECENTO BALLATE HEATHER HOLMQUEST

To perform early music, we must reconcile two desires: one, to inform ourselves of relevant scholarly work that pertains to the music in question, and two, to make our performance accessible to modern audiences. Adding to this is the two-fold problem of not having enough sources to paint a complete picture of what the music would sound like, and also having the benefit and/or curse of a twenty-first century musical ear both on the part of the audience and the performer. Thus, the question is, are these desires for accurate historical representation and a complete musical experience inherently mutually exclusive, or can we find some middle ground?1 Can we reach that middle ground without being charged with over-speculation? To put it another way, when performing early music, there is always guesswork to be done. How do we get better at guessing? My goal is to show that one can use some forms of modern musical analysis to both illustrate the unique features of early music, as well as highlight any commonalities that early music has with its descendants in common practice and beyond. I do this by highlighting structures and formal elements in the music that, when performed with these things in mind, it creates a sense of familiarity in the listener, and builds up a set of musical expectations that audiences may lack when first listening to early music. To explore the intersection between analysis, historical musicology, and performance, I focus on the monophonic repertories of the Squarcialupi and Rossi codices.2 The Rossi Codex, compiled c. 1370, is one of the most extensive early Trecento sources.3 The Codex portions that we have contain 37 secular works, five of which are monophonic ballate.4 The Squarcialupi Codex is the largest compiled source of Trecento music, and contains seventeen works by Lorenzo da Firenze and sixteen by Gherardello da Firenze.5 Each of these composers contributed five monophonic ballate to

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the Codex. I’ll be discussing three works, two from the Rossi manuscript (“Che ti zova, nascondere,” and “Amor, mi fa cantar la francescha,” both by anonymous composers) and one from the Squarcialupi Codex, “I vo’ bene a chi vol bene,” written by Gherardello da Firenze.6 While “Che ti zova” exhibits some features of tonal structure and linked ripresa and piedi sections, “Amor mi fa cantar” is more irregular.7 It is “I vo’ bene” that shows full cyclicity, as I describe below, and I suggest that as a composition from the Squarcialupi Codex, this means that it is more structurally organized than the compositions from the Rossi Codex. While composers and theorists in the Trecento period may not have conceptualized or discussed their music in the same terms that we modern listeners conceptualize tonal music, there is an undeniable continuity, or structure, to modal music that modern analysis uncovers and modern listeners find familiar. Continuity can be created in a number of ways: form, motive, and pitch center, to name a few. In performing Trecento music, I have found that it is useful to determine pitch centers, highlight the form in a formes fixes piece, and bring out the contrapuntal structure. In polyphonic music, this task is aided by the conflict and resolution present in the dissonances and consonances between two or more voices. In a monophonic song, however, the task is more convoluted. Are there consonances and dissonances present when only one voice is singing? I assert that a single line melody, whether modal or tonal, creates a permeating structure that points the listener towards a pitch center, whether we call that pitch the finalis or the tonic. When I refer to structure, I do not mean the Ursatz, or fundamental structure, of an unfolded tonic triad; obviously, this music is not built on triads. It is, however, organized into descending step-wise lines that span intervals of thirds, fourths, and fifths. It also employs cadential figures, and emphasizes pitch-spaces that can be labeled with general final, dominant, and subdominant areas. I have developed a version of voice-leading analysis to show the internal logic of each individual song, thus allowing me to find similarities between pieces, as well as departures that this music makes from traditional tonal strategies. In Harmonielehre, Heinrich Schenker discounts the modal system as “most inappropriate for the development of motivic intentions,” adding, “or, at any rate, …[it] would engender situations far too unnatural for any style to cope with.”8 However, I consider some of Schenker’s techniques for voice-leading analysis to be in fact appropriate for the purposes of showing voice-leading patterns in fourteenth century ballate. I am not the first to do something like this; in fact, Felix Salzer used Schenker’s methodology to attempt to chart the development of tonality in Western music. He was arguably successful at pointing out

Structural Cyclicity in Trecento Ballate

23

features of voice-leading in the music of Leonin and Pérotin, however, his agenda to uncover the origin of tonality led to some unsuccessful 9 interpretations of medieval counterpoint. More recent scholars such as Daniel Leech-Wilkinson and Cristle Collins Judd have also explored voice-leading analysis of early music.10 Leech-Wilkinson began his analysis of Machaut with a criticism of Salzer’s agenda to seek tonality in early music, and produced an analysis of Machaut’s “Rose, Lis.”11 Cristle Collins Judd provides an excellent example of structural analysis in her article, “Some Problems of Pre-Baroque Analysis: An Examination of Josquin’s Ave Maria … Virgo Serena.”12 She reflects on the “emerging tonality” in works of the fifteenth century, and considers structural analysis of early music valuable “since the twentieth-century observer perceives both tonal and modal elements.”13 As I explored the structures of the monophonic ballate in these two sources, I have been aware of some key differences between pre-tonal and tonal melodies. The changing metric organization made structural analysis more complicated than when investigating a tonal work. I adjusted my analysis by taking into account the specific pitches highlighted by the text placement in the manuscripts. Secondly, pre-tonal melodies are highly step-wise. They tend to be decorations of step-wise structures that descend from a fifth above a finalis. I have been careful not to force these pieces into a particular concept of a 3- or 5-line Urlinie as described by Schenker, however; when a piece is irregular, and spans a fourth at the background level, I let it exist as a fourth. We will see an example of this in “Amor, mi fa cantar a la francesca.” The most interesting feature, however, is that the structure of these pieces aligns with, and reinforces, the repetitious poetic and musical scheme of the ballata form. The presentation of the ripresa, journey to the piedi, and return back to the ripresa is not just cohesive on a melodic level (for example, the intervals between each sections are often fifths or octaves, and emphasize tones related to the finalis), but on a structural level as well. This formal structure is the focus of this study; I wish to highlight the development of structural cyclicity, that is, structure that creates an entirely closed system of descending pitches in the monophonic songs of the Trecento period. Nearly all ballate found in the Rossi and Squarcialupi codices elaborate a  structure in the ripresa, however, the largest source of variety is how the piedi relate to the ripresa. By examining the structural underpinnings of these songs, I find that despite longer phrases and displaced metric motion, structural motion is more organized in the Squarcialupi codex than in the Rossi codex. The different type of piedi

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Chapter Two

structures are outlined below, followed by a table that classifies each piedi type by name, source, and overall structure. 1. Modal modulation. The piedi section occupies a different mode than the ripresa, completing a 5-line structure in the new mode. The mode of the piedi can be related by a fifth (sounding akin to a binary structure in tonal music), a fourth, or a second above or below the actual finalis of the ripresa. “Che ti zova nascondere,” one of the pieces examined here, features a ripresa with a 5-line structure from A to D, followed by a piedi section that modulates to the subdominant by emphasizing a 3-line structure from Bb-G. 2. Sectional interdependence. The piedi section requires the beginning of the ripresa to form a complete structural line in the piedi. A highly irregular work from the Rossi codex, “Amor mi fa cantar,” will be presented shortly. It contains a 4-line structure that arcs from D at the beginning of the ripresa to A in the piedi. This results in an inconclusive finalis of C at the end of the ripresa, despite the piece’s emphasis on D and A. 2a. Full cyclicity. This is a subtype of sectional interdependence where a full octave is present in the background of the piece, beginning at the top of the piedi, and finishing with the finalis of the ripresa. Some pieces exhibit a behavior that I call “full cyclicity,” defined as such: the song starts on , descends to , and then the piedi follows a  descent in the same mode. There are five pieces that do so in the monophonic ballate in the Squarcialupi codex that I examined for my dissertation, and six that do not. Interestingly, of these five pieces, four of them have a modal center on G. As shown in Table 2-1 below, there are seven pieces that are centered in G, and four of them, 57%, are cyclic. “I vo’ bene a chi vol bene,” the last piece presented in this article, demonstrates this structure. 3. Dominant interruption. This describes when the piedi concludes in an interruption, as in,  of a 5-line descent. It can either conclude with the corresponding  at the beginning of the ripresa, or 
a fourth above that concluding pitch.

D-C

A-D

F#-D

G-C

D-G

D-A

A-D

D-G D-G

D-G

A-F

A-D A-D

C-F

D-G

Amor mi fa cantar

Che ti zova nascondere

Lucente stella

Non formò Cristi

Per tropo fede

De', poni, amor a me

Donna, l'altrui amor

I’ vivo amando I' vo' bene

Per non far lieto alcun

Non vedi tu, amore

Non so qual i' mi voglia Non perch'i' speri

Donne, e' fu credenza

Sento d'amor la fiamma

Rossi

Anonymous

B -G

G-D

F-C

D-G A-E

A-E

G-D

Lorenzo

Lorenzo

Lorenzo Lorenzo

Lorenzo

Gherardello

Gherardello Gherardello

Gherardello

D-G G-C G-D

Gherardello

Anonymous

Anonymous

D-B

D-A

C-D

G-C

Squarcialupi

Squarcialupi

Squarcialupi Squarcialupi

Squarcialupi

Squarcialupi

Squarcialupi Squarcialupi

Squarcialupi

Squarcialupi

Rossi

Rossi

Rossi

Rossi

Anonymous

B -A

E Anonymous

Codex

Composer

Piedi

E

Table 2-1, Piedi types in the Rossi and Squarcialupi Codices

Ripresa

Name

Full Cyclicity

Full Cyclicity

Subdominant Modulation Interruption

Interruption

Full Cyclicity

Dominant Modulation Full Cyclicity

Interdependence/Modulation

Interdependence

Interruption

Interruption

Subtonic Modulation

Subdominant Modulation

Interdependence/Interruption

Piedi Type

Structural Cyclicity in Trecento Ballate 25

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Chapter Two

“Che ti zova nascondere” Among the ballate in the Rossi codex, "Che ti zova nascondere" exhibits structure that is significantly more "proto-tonal" within each individual section of music. However, it is a ballata minima, so the shorter phrases and sections do not grow to be as complex as the pieces I will investigate later in this study. The piedi section relates to the ripresa, in D, by way of modal modulation to the subdominant key of G. There may be a textual reason for this; in the ripresa, the narrator is calling for his beloved to come out from hiding, both literally (to reveal her face) and figuratively (to profess her love for him).14 In the piedi, she remains hidden, obfuscated, and this is indicated by a number of exclamations: she’s a jewel of unknown worth, hidden from view, and remains so until his desire can no longer be held in check. In the ripresa and volta, the speaker’s feelings come out into the open. This is an example of how the two sections are differentiated both by meaning and music. Thus, the journey from the pitch center, D, to the subdominant, local pitch area of G is warranted by this juxtaposition. Another feature of the text is found in the title; a common practice in the Trecento period is to embed the names of women, perhaps as dedications, in the poetry. In the two songs I discuss from the Rossi Codex, they both appear in the ripresa. In “Che ti zova nascondere,” the song is about “Giovanna,” presumably the woman who hides her face. Che ti zova nascondere'l bel volto?

Why would you wish to hide your lovely face?

A. Che ti zova nascondere'l bel volto?

Why would you wish to hide your lovely face?

b. Donna, la bella pietra, stando ascosa, b. Nessun puo dir quanto sia preciosa; a. Ma chi la vede, si la loda molto.

Lady, no one can tell how much a gem is worth, if it is hidden from view; But when it is in view, it is much praised.

b. Cum più t'ascondi, più desio mi mena; The more you hide, the more desire torments me; b. Donca non voler più ch'io porti pena, Do not, therefore, prolong my suffering, a. Ch'amor per ti servir lo cor m'à tolto for love has taken my heart and placed it in your service.

The voice-leading analysis of the ripresa, as seen in Example 2-1, is structurally straightforward and projects the finalis of D from the opening descent. The opening gesture in mm. 1-4 is a prefix; it consists of a descending line from D to A, similar to other songs in this repertoire. I call

Example 2-1, “Che ti zova nascondere,” voice-leading analysis of the ripresa

Structural Cyclicity in Trecento Ballate 27

Structural Cyclicity in Trecento Ballate

29

it a prefix because the leap from D to A in m. 2 suggests two voice-leading tracks (an upper and lower voice). Since the first four measures are followed by consistent elaborations of A in mm. 4-8, I treat A as the structural pitch that sustains throughout the ripresa until the descent in mm. 9-11. As a result, the background structure of the ripresa is a 5-line descent from A to D. The ripresa behaves much like a section of a tonal piece, with a 5-line overall structure (i.e. descending structural line consisting of five pitches, terminating in the finalis) that descends to a cadence that marks a formal boundary. The piedi section of “Che ti zova nascondere,” shown in Example 2-2, clearly projects a tonality that departs from the ripresa, tonicizing G by way of pivoting on the A in mm. 12-15. The section begins with an E and ascends directly from the end of the ripresa, creating an ascending passage from the D finalis to the A structural tone, or what seems to be the prevailing structural tone. This A in m. 15, in turn, acts as a pivotal
for the modulation of the piedi to G in m. 18. The modulation to G is confirmed when the melody cadences once again in m. 22 with a stronger cadential formula, including a written F#, and structural descent from Bb to G. The ascent continues in the second half of the piedi to a C, not quite reaching the upper D that could be found in a cyclic ballata, and I treat it thus as an upper neighbor to a Bb. The music then descends to a cadence on G with a supportive F# leading tone written in the manuscript. This piedi section can be viewed as a development section that tends toward the 'subdominant;' it spans from a step above the finalis to a step below the finalis and cadences on G, a step below the dominant. The re-orientation for the performer back to a D is thus made easier because the starting pitch for the ripresa is  in G. Thus, the piedi section, tonicizing G, is a smooth structural link back to the A. The link between the piedi section and the ripresa is shown in the background analysis, shown below in Example 2-3.

Example 2-3, “Che ti zova, nascondere” background analysis

“Amor mi fa cantar” Voice-leading is often obscured in the Rossi Codex, even in relatively short pieces, such as “Amor mi fa cantar.” A cursory glance shows that

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each section is made of two short phrases, and the beginning and ending points of the two sections taken together form a descending scale: D-CBb-A. If I were to impose an Urlinie on this piece, or, rather, make this piece conform to the expectations of a typical Schenkerian graph, a single key area would be determined, and the overall structure would have to conform to that key area. Instead, I would like to try to explain this piece from a voice-leading perspective instead, for the sake of demonstrating a lack of tonal coherence. The text displays a typical trope shared among both French and Italian poets: a man loves a lady so well that he would die as a result of his torment, but because he fears her rejection, he does not tell her of his affections. The text and translation can be found below. The lines of poetry have one complete thought beginning at the top of the piedi and finishing with the volta in each respective verse. Thus one might expect, musically, that the melody outlines a complete structure that begins with the piedi and concludes with the ripresa, but instead, a 4-line structure descending from D to A begins in the ripresa and concludes in the piedi section, placing a cadence on C at the close of the ripresa. Amor mi fa cantar

Love makes me sing

A. Amor mi fa cantar a la francesca.

Love makes me sing in the French style.

b. Perché questo m'aven non olso dire. b. Ché quella donna che me fa languire languish a. Temo che non verebe a la mia tresca.

Why this happens to me I dare not say. For I fear that the one who makes me

b. A lei sum fermo celar el mio core b. E consumarmi inançi per so amore. a. Ch'almen morò per cosa gentilesca.

I am resolved to hide my heart from her And rather to waste away for her love. So that at least I die for a noble thing.

b. Donne, di vero dir ve posso tanto, b. Che questa donna, per cui piango e canto a. È come rosa in spin morbida e fresca.

In truth, o ladies, I can tell you this much, As this lady for whom I weep and sing

Would not come to my dance.

Is soft and fresh as a rose in thorns.

The ripresa and piedi sections together form a descent from D to A, which is an augmentation of the opening descent from D to A in mm. 1-3. These two sections together project the top half of the descending melodic minor scale. With the exception of the very last note in the ripresa, the DA structure fits the music. If D were the finalis of this piece, it would also support the conclusion of the piedi on the A, and bring out the Bb as highly dissonant, requiring the resolution to the A and the cadence in m.

Structural Cyclicity in Trecento Ballate

31

13. These two sections together project the top half of the descending melodic minor scale, shown in the background summary below in Example 2-4.

Example 2-4, “Amor mi fa cantar” background analysis

How, then, do we explain the final cadence? The finalis of the piece is a C, indicating that the overall structure should outline a 3-line descent from E, or a 5-line descent from G in a tonal reading. The pitches form a cadential figure that circles around C as well. However, there is no upper E in the entire piece, thus the piece lacks even a structural . This results in a type of structural organization in the piece that relies on an interconnected sense of melodic line leading from the ripresa to the piedi, rather than from the piedi to the ripresa. We might consider this piece as further away from the development of tonality than the other pieces in the Rossi codex, and certainly less tonal than those in the Squarcialupi codex, but again, it is more important to find out what does make this piece cohere. To reconcile the two pitch centers, D and C, I indicate in my voiceleading graph, Example 2-5, that instead of having a structure based on thirds, this piece outlines intervals of a fourth and fifth. If C is the destination of the underlying structure, we can view mm. 5-7 as an ascent from G to C, while mm. 1-4 form a descent of a fourth from D to A. Thus, the fourths are unfolded to avoid parallels, and the opening D is the structural pitch that binds the two phrases together. This creates a fourthfifth-fourth chain of intervals, which is a departure from the common chains of thirds that Trecento monophonic pieces often display. Assuming that “Amor mi fa cantar” centers on C, it would make some sense that the piedi, with the emphasis on Bb and A, could be analyzed in G, creating a dominant function. The F# in the manuscript adds weight to this argument, and thus the piedi section concludes with an interruption. The Bb and A act as a temporary  and , respectively. The piedi section is more typically third-governed, which is established by the passing tone figures in m. 9 between G and Bb, and in m. 10 between A and F.

Structural Cyclicity in Trecento Ballate

33

Another reading of “Amor mi fa cantar” is that of a descending fifths progression. If the first phrase that outlines D-A is treated as a harmonization of D, then the second phrase harmonizes G (the D voice in the background forms a fifth, reinforcing a descending fifths scheme. Finally, the G acts as a dominant to C, the finalis of the piece and the resolution of the D in the upper voice. The piedi serves more than one purpose, then: the auxiliary key of G-minor is akin to modulating to the dominant in a tonal piece, and the interrupted ending sounds like ending on the dominant chord of G-minor, that is, a D chord. This D chord then acts as the initial sonority of the ripresa, essentially modulating from V/Gminor to I/D-minor.

“I’ vo’ bene a chi vol bene” “I’ vo’ bene” demonstrates what I refer to as a fully cyclic form; the 5line in the ripresa is extended to a full octave by the 


 structure of the piedi. The text, with translation provided below, involves many plays on words that are amplified by the cyclic form of the ballata. The speaker in the poem discusses the reciprocation of love, and insists that he only loves those who love him: in effect, he requires the lady to show him love first before committing his love to her. This poem is in effect a giant loop of affection, giving it, taking it, and sorting out feelings amongst two people. The cyclic structure supports this idea, seamlessly moving from the ripresa to the piedi via the same tonal center.15 I’ vo’ bene

I love the one

A. I' vo' bene a chi voi bene a me E non amo, chi ama proprio sé.

I love the one who loves me And I do not love one who only loves herself.

b. Non son colui che per pigliar la luna Consuma'l tempo suo e nulla n'a; b. Ma, se m'avvien ch'amo m'incontri d'una Che mi si volga, I' dico – E tu ti sta!

I am not one who tries to seize the moon, Wasting my time and ending up with nothing. But, if love brings me a lady

a. Se me fa: – Lima, lima! – et io a lei: Dà, dà! – E così vivo in questa pura fe'.

Who turns me down, I say – You're on your own! If she says: – Take me, take me! – I say to her Give me, give me! – Thus I live with this simple faith.

b. Com' altri in me, cosi mi sto in altrui, As others are to me, so I am to other people, Di quel ch’i’ posso, a chi mi dona do. Of those who give to me, I give them what I can. b. Niuno può dir di me: vedi colui, No one can say about me: look at him,

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Che con duo lingue dice, si e no.

With two tongues he says both yes and no.

a. Ma fermo a chi sta fermo sempre sto;

But I stand firm with those who stand with me. If they serve my needs, I will serve theirs.

S'io l'ho al bisogno mio, me à a sè.

The piece is centered on G, and it displays features of G Dorian; this is indicated by the immediate leap from G to D at the beginning, the flat in the key signature, and the conclusion of the ripresa on a G, as seen in my voice-leading analysis of the ripresa in Example 2-6. The ripresa generally projects a 5-line progression from D to G as well. The opening phrase of “I’ vo’ bene” prolongs the D while the inner voice outlines a 

 pattern, that is, G to A to Bb. The next phrase, mm. 7-10, is a neighboring area that explores the notes between C and F. The F is not a “chordal” skip per se, since we have no chords here, but it could be considered a consonant skip. The reach to F foreshadows the high point of the piedi section, where the F is emphasized. The C is a lower neighbor to the prolonged D that shows up again in the conclusion of the phrase in m. 11. The next passage, mm. 11-16, is predominantly a melisma on the “o” of “propio;” this is a melodic sequence that drives to the cadence and initiates the descent of the 5-line progression. The last two measures are a truncated version of the motive in the sequence that pairs  with  right before the resolution to G. The ripresa concludes with a cadence on G, and the leading tone, F#, is in the manuscript. In my transcription shown on p. 15, the F# appears in m. 14 and remains active through m. 15. The octave leap between the ripresa and piedi between mm. 16-17 allows for another descending line, because descending further from the low G would get rather low for any vocalist. The piedi’s structure, shown in Example 2-7, is a descending line from G to D, with an inner voice in thirds accompanying it. This section is rather straightforward, although the conclusion of the phrase in mm. 20-24 involves E and C. The transcription of this piece that I worked with suggests a C#, but it happens so quickly and is not a typical cadential gesture, so it is unwarranted.16 The structure of this ballata “links up,” so that not only are the melodic junctures easily navigated (the octave leap in between the ripresa and piedi, and the D leading back to the G-D duality from the piedi back to the ripresa), but from the ripresa to the piedi, there is one continuous structural line from D down to D, or 
, with an octave transfer in the middle. This can be seen in the background summary of the form, shown in Example 28. Thus, this piece indicates a move towards the practice of single chord prolongation used in the common practice period.

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Example 2-8, “I’ vo’ bene,” background analysis

In this study, I hope to have shown some basic structures found in the Rossi and Squarcialupi Codices, though I acknowledge more analysis is needed to completely describe the interactions between structure, mode, text, and rhythmic mode. Some of the other lines of inquiry to take include the relationship of meter to the parsing of modal and cadential expectations, the placement of syllables in relation to structure, and the comparison of motivic and structural patterns in these songs to other early music works in an effort to redefine the link between modal and tonal music. Finally, it is my intention to continue my investigation of modal structure in the polyphonic works that accompany these monophonic songs.

Notes 1

Nick Wilson, The Art of Re-Enchantment: Making Early Music in the Modern Age (New York: Oxford University Press, 2014), 191, weighs in on this question in his study of early music performance in Britain. He emphasizes that while performers of medieval music in some sense need to be scholars, they also need to be musicians first, which entails bringing life to the music. 2 Monophonic works of the fourteenth century generally occupy a subordinate place in scholarship, but this should not be the case. For example, Susan Rankin in “Some Medieval Songs,” Early Music 31, no 3 (2003): 330, asks, “But what of monophonic songs? In as much as polyphony features in modern conceptions of Parisian musical culture of the early thirteenth century, so monophony has remained largely outside the knowledge and experience of singers and scholars.” She goes on to write, “in the various layers of Parisian polyphonic repertories new techniques of music composition…were being explored, concern with expression of words in music was losing prominence.” 3 The facsimile and commentary of the Rossi codex is compiled by Nino Pirrotta, ed. in Il Codice Rossi 215: Studio introduttivo ed edizione in facsimile (The Rossi codex 215: Introductory Study and Facsimile Edition) (Lucca: Libreria Musicale Italiana, 1992). 4 Ballate follow the form AbbaA, which is similar to the French virelai. 5 The beautiful Squarcialupi facsimile is edited by F. Alberto Gallo, Il Codice Squarcialupi: Ms. Mediceo Palatino 87, Biblioteca Laurenziana Di Firenze (Venice: Giunti Barbèra, 1992).

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Full analyses of the fifteen monophonic ballate in the Rossi and Squarcialupi codices can be found in my dissertation: Heather Holmquest, “Structure, Musical Forces, and Musica Ficta in Fourteenth-Century Monophonic Song,” PhD diss., University of Oregon, 2014). 7 The ripresa is the refrain, and the piedi are the verses that are set to contrasting melodic material. The volta of a ballata is musically equivalent to the ripresa. After the first use of these terms, I have elected not to continue using them in italics. 8 Heinrich Schenker, Harmony, ed. Oswald Jonas, trans. Elisabeth Mann Borgese (Chicago: Chicago University Press, 1954, rpt. 1980), 56. 9 Felix Salzer, “Tonality in Early Medieval Polyphony: Towards a History of Tonality,” The Music Forum, Vol. 1, ed. William J Mitchell and Felix Salzer (New York: Columbia University Press, 1967), 35-98. 10 Felix Salzer’s work with structural analysis is summarized in John Koslovsky’s recent dissertation, “From Sinn und Wesen to Structural Hearing: The Development of Felix Salzer's Ideas in Interwar Vienna and Their Transmission in Postwar United States” (PhD diss., Eastman School of Music, University of Rochester, 2009), 163-166. 11 Daniel Leech-Wilkinson, “Machaut's ‘Rose, Lis’ and the Problem of Early Music Analysis,” Music Analysis 3, no. 1 (1984): 14-17. 12 Cristle Collins Judd, “Some Problems of Pre-Baroque Analysis: An Examination of Josquin’s Ave Maria … Virgo Serena,” Music Analysis 4, no. 3 (1985): 222-23. 13 Ibid., 223. 14 I have consulted and modified translations written by Giovanni Carsaniga, found in “Rome, Biblioteca Apostolica Vaticana, Rossi 215,” La Trobe University Medieval Music Database, last modified March 5, 2003. http://www.lib.latrobe.edu.au/MMDB/Mss/RS.htm. 15 This English translation is modified from William Hudson’s DMA document, “Performing Music of the Trecento: A Case to Rethink Our Modern Editions,” DMA Thesis (Indiana University, 2012), 51. 16 Johannes Wolf, Der Squarcialupi-codex, Pal. 87 der Biblioteca Medicea Laurenziana zu Florenz. Zwei- und dreistimmige italiensche weltliche Lieder, Ballate, Madrigali und Casse des vierzehnten Jahrhunderts (The Squarcialupi codex, Pal. 87 from the Medicea Laurenziana Library in Florence. Two- and Three-Part Traditional Italian Secular Songs, Ballate, Madrigals and Caccie of the Fourteenth Century) (Lippstadt: Fr. Kistner & C.F.W. Siegel, 1955), 57.

CHAPTER THREE MOZART’S COMMON (YET UNCOMMON) COMMON-TONE TRANSFERS SUSAN K. DE GHIZÉ

Pivot chords exemplify the idea of duality: a chord functions in one way in the first key and another way in the second key, resulting in dual meaning. Similarly, single tones can have more than one interpretation. Historically, only one type of “pivot tone” has been emphasized—that from the common tone modulation. Most textbooks define a common-tone modulation as having three conditions: 1) both keys (or chords) have only one tone in common, and the common tone is singled out to connect two keys (or chords); 2) both keys (or chords) are either major or minor; and 3) when they are both major, they are related by a major third; or when they are both minor, they are related by a minor third, resulting in a chromatic mediant relationship. Common-tone modulations are most frequently found in nineteenth-century literature, by such composers as Schubert and Liszt. Despite its most popular usage, a common tone can technically be allowed to connect any two keys (or chords). The quantity of notes in common, the quality of both chords, and the relationship between the two keys (or chords) should not matter. If this is the case, thirty years before Schubert, Mozart frequently incorporated common tones. In order to distinguish these less-traditional utilizations from conventional commontone modulations, I use the term “common-tone transfers.” 1 Indeed, although Mozart exploited common tones between chords, they were by no means used only as an instrument for modulation. Also, the designation “common-tone transfer” underscores the idea of dualism, as one note in common shifts in meaning. In this paper, I show that in his piano sonatas, Mozart applies common-tone transfers in a variety of ways. Common-tone transfers occur most often between sections (and sometimes between movements or even entire sonatas). In movements that are in sonata form, common-tone transfers happen most frequently

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between the main and subordinate themes or between the exposition and development. When a common-tone transfer is used in the exposition, Mozart will sometimes capitalize on this technique in the development. It is remarkable how often Mozart cadences on an unaccompanied pitch, or begins a theme singling out one note. A survey of Mozart’s piano sonatas shows that there are no less than 50 instances in which he uses common-tone transfers. I label the specific types of common-tone transfers by the position of the note in the first chord (R for root; T for third; F for fifth; and S for seventh) followed by the position of the note in the second chord (again, R, T, F, or S). For example, the most common usage occurs when the root of a major triad becomes the fifth of another major triad; this would be called an RF (root-fifth) common-tone transfer, or simply, RF. A model illustration of an RF is in the first movement of the Piano Sonata No. 7 in C major (K. 309). In Example 3-1a, the transition after the main theme ends with the unaccompanied pitch D at bar 32, repeated three times, and acting as the root of a D major triad. The subordinate theme also emphasizes the single pitch D, but with lower neighbors. When the full harmony does arrive at bar 35, we find that the music is actually in G major, and not in D major. Since the root of the first harmony (D in D major) becomes the fifth of the second harmony (D in G major), this results in an RF. In Example 3-1b, Mozart repeats the pattern but ends on G major instead of D major, stressing the unaccompanied pitch G. Having already heard this scheme in the exposition, the listener is not surprised to hear another RF in the recapitulation.

Example 3-1a, K. 309/i, mm. 31-35

Example 3-1b, K. 309/i, mm. 125-129

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On the other hand, Mozart does surprise us in the recapitulation of the first movement of the Piano Sonata No. 5 in G Major (K. 283). In Example 3-2a, the main theme in the exposition finishes at bar 22 with an unaccompanied pitch D, which is the root of a D major triad. The subordinate theme begins in bar 23, also in the key of D major. Accordingly, no transfer or modulation takes place here. Example 3-2b shows the main theme area of the recapitulation, which is exactly the same as the exposition, and ends on the single pitch D at bar 89. However, this time, the subordinate theme begins in G major in bar 90, giving the D the double meaning of root of D major, and fifth of G major: an RF. By merely comparing Examples 3-1a and b to 3-2a and b, we can already predict Mozart’s unpredictability.

Example 3-2a, K. 283/i, mm. 21-24

Example 3-2b, K. 283/i, mm. 88-91

Mozart uses a similar technique in the third movement of this same sonata, found in Examples 3-3a and b. The main theme ends with a D major triad in the exposition (m. 40) and recapitulation (m. 211). When the subordinate theme begins in the exposition, Mozart singles out A—the fifth of the chord—with lower and upper neighbors decorating it. The first supporting harmony at bar 43 is a D major triad, which results in no common tone transfer. In the recapitulation, rather than singling out A, Mozart singles out D, which is the root of the D major triad. Lower and upper neighbors decorate the D, which becomes the fifth of a G major triad (RF). Although the main themes of the exposition and recapitulation are the same, Mozart uses an RF to return to the home key in the recapitulation. Unlike Example 3-2, where the unaccompanied pitch

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concluded the main theme, Example 3-3 makes use of the single pitch at the start of the subordinate theme.

Example 3-3a, K. 283/iii, mm. 39-43

Example 3-3b, K. 283/iii, mm. 210-212

RFs are not limited to movement from one major triad to another, although that is the most frequently occurring category. Mozart also applies common-tone transfers from major chords to minor chords using RFs. Example 3-4, taken from the second movement of the Piano Sonata No. 2 in F major (K. 280), displays the end of the development and what appears to be the start of the recapitulation. The development ends in bar 32 with the solitary pitch G, which is the root of a G major triad. At bar 33, the main theme reappears in C minor. The single pitch G connects the harmonies of G major to C minor, making this an RF.

Example 3-4, K. 280/ii, mm. 31-34

Skeptics might be unconvinced at this point, claiming that Mozart is simply returning to the home key in the recapitulation, and this is almost inevitably accomplished by dominant at the end of the development.

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However, although Mozart frequently used RFs to modulate from the dominant back to the home key, he rarely used common tones to modulate from a chord to its dominant, or FR, although they also share one common tone.2 Mozart does make use of FR common-tone transfers in his piano sonatas, albeit infrequently.3 RFs do not have to be from one triad to another. In Example 3-5, which comes from the second movement of the Piano Sonata No. 4 in EFlat Major (K. 282), the A section of the minuet begins in B-flat major, but modulates to the dominant, F major. The A section concludes with the lone pitch, F, in bar 12. The B section retains the pitch F in the bass clef, but Mozart harmonizes the F in the treble clef with a B fully diminished seventh chord, giving F the double meaning as root and fifth.4

Example 3-5, K. 282/ii, mm. 11-13

Second to RFs, Mozart uses the RR (root-root) common-tone transfer most often. This is when the root of a major triad becomes the root of a minor triad (or vice versa). One may question this as a type of commontone modulation, since there are two notes in common: the root and the fifth. But RRs only occur when a solitary common tone, which in this case is the root of both chords, is set apart. Others may question the lack of double meaning, as the root of a chord remains the root of a chord. However, since the root of a major triad becomes the root of a minor triad, these are still RRs since they have changed meaning. Almost all RR common-tone transfers occur between the end of the exposition and the start of the development, since the stark unexpectedness of the parallel mode is fitting for the appearance of the development. One case can be found in Example 3-6. The exposition ends in A major, with a full rolled chord at bar 51. The development begins in bar 52 with the pitch A singled out, followed by a chromatic lower neighbor. On beat two, C-natural confirms that we are no longer in A major, but rather, A minor. Once again, Mozart sets apart the root of the chord to modulate.

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Example 3-6, K. 284/i, mm. 50-52

An additional type of RR common-tone transfer actually contains three common tones, although only one is utilized. Mozart will occasionally convert a major triad into a major-minor seventh chord, making it the dominant seventh of the following chord. He uses this type of RR in the third movement of the same sonata, shown in Examples 3-7a and b. In order to understand Mozart’s deliberateness of this type of transfer, one can compare what happens in the exposition to what happens in the recapitulation, similar to the other analyses with parts a and b. The main theme in both the exposition and recapitulation ends exactly the same way in bars 10 and 96: that is, on the unaccompanied pitch C, which is the root of a C major triad. In the exposition, the transition begins with a C major triad; however, in the recapitulation, the transition begins with a C majorminor seventh chord. Since there is no transfer in Example 3-7a, this is not an RR. Contrariwise, the function of C clearly differs in Example 3-7b, as the root of a major triad transforms into the root of a dominant seventh chord.

Example 3-7a, K. 279/iii, mm. 9-12

Example 3-7b, K. 279/iii, mm. 95-98

In addition to RRs, FFs (fifth-fifth) can also be found in Mozart’s piano sonatas, as illustrated in Examples 3-8 and 3-9, both from K. 280. In

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Example 3-8, the development of the first movement begins in C major. The common tones in this example show how the term common-tone transfer is more appropriate than common-tone modulation. On the downbeat of bar 58, the unaccompanied pitch G has double meaning as the fifth of a C major triad and the fifth of a C-sharp diminished triad, resulting in an FF. Similarly, another FF occurs on the downbeat of bar 60, where D functions as the fifth of a G major triad and a G major-minor seventh chord.

Example 3-8, K. 280/i, mm. 57-61

Example 3-9 comes from the second movement of K. 280, which begins in F minor. The exposition ends at bar 24 in the relative major, Aflat major, with doubled root and a third: it is lacking the fifth. The development begins with this fifth: the single pitch E-flat, which is decorated with an upper neighbor. The harmony that finally arrives at bar 26 is not A-flat major, however; indeed, it is not even A-flat minor or Eflat major. Rather, E-flat becomes the fifth of an A fully diminished seventh chord. This FF common-tone transfer is quite unexpected, but as it is the start of the development, it is expectedly unexpected.

Example 3-9, K. 280/ii, mm. 23-26

Notice that when Mozart uses RRs, he changes the quality of the triads from major to minor. On the contrary, when he uses FFs, he alters the quality of the triads from major to diminished. He does not use FFs to progress from a major chord to its parallel minor. Moreover, Mozart never uses TT common-tone transfers in his piano sonatas. Another example of the difference between common-tone transfers and common-tone modulations can be found in Example 3-10, which comes

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from the second movement of K. 284. C-sharps abound in bars 47-49. Although the C-sharps never appear by themselves, the concept of a common tone connecting these bars is clear. Mozart harmonizes the first group with an F-sharp minor triad, the second group with a C-sharp major-minor seventh chord, and the last group with an A-sharp diminished seventh chord. Each occurrence not only creates more dissonance and tension, but also changes the position of the pitch C-sharp. As the C-sharp has triple meaning, this would be an example of an FRT (fifth-root-third).

Example 3-10, K. 284/ii, mm. 47-49

There are a limited number of examples in which Mozart uses the “traditional” 19th-century common tone modulation that was described in the introduction.5 An RT (root-third) progressive modulation can be found in Example 3-11, which comes from the second movement of K. 283. Mozart modulates from an E major triad to a C major triad, taking advantage of their common tone, E. The development ends in bar 23 on the single pitch E, tripled, acting as the root of an E major triad. The recapitulation has a chromatic anacrusis that leads us to a C major triad in bar 24. As a result, this is not only a common-tone modulation as described by textbooks, but also an RT.

Example 3-11, K. 283/ii, mm. 22-24

Mozart uses another traditional common-tone modulation at the end of the development from the first movement of his Third Piano Sonata (K. 281). But unlike Example 3-11, Example 3-12 is an FS rather than an RT. At bar 66, the harmony is an E-flat major triad. The melody then ascends to B-flat, which is repeated four times with lower neighbors. On the last

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eighth beat of bar 67, the B-flat becomes the seventh of a C major-minor seventh chord, thus making this an FS.

Example 3-12, K. 281/i, mm. 66-68

An E-flat major triad and a C major-minor seventh chord have two common tones: not only does the fifth become the seventh, but the third can also become the fifth. However, just as Mozart freely used RFs, but infrequently FRs, he also applies FSs numerous times in his piano sonatas, but seldom TFs.6 One of those seldom-used examples of a TF (third-fifth) can be found in Example 3-13, which comes from the third movement of the same piano sonata. The movement is a rondo and at bar 51, the music modulates from B-flat major to G minor. There are two notes in common between B-flat major and G minor: B-flat and D. Mozart ends the main theme section with the pitch B-flat tripled. A single line leads into the next couplet. The last single pitch in bar 51 is D and that lone D is struck again to start the couplet, which begins in G minor. Mozart had two choices for utilizing common tones in this example: B-flat or D. One could argue that B-flat is a strong candidate for that common tone, but in my opinion, the D is what most effectively connects the two sections, and the two keys.

Example 3-13, K. 281/iii, mm. 50-53

*** Having all these different types of common-tone transfers at his fingertips, Mozart creates surprises, as well as unfulfilled expectations. Mozart makes the most of common tones in his First Piano Sonata in C major (K. 279). Indeed, nearly every transition linking main and

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subordinate themes, every move from exposition to development, and every return to the recapitulation utilizes common tones. In the first movement, we hear an FS connecting the first and second theme in the exposition, which is shown in Example 3-14a. The main theme closes with a G major triad; the subordinate theme begins with the pitch D, which we expect to be the fifth of a G major triad or the root of a D major triad, since the movement is in C major. Instead, Mozart transforms this D into the seventh of an E major-minor seventh chord. The subordinate theme is a sentence, and Mozart duplicates the twomeasure idea in sequence. This time, the sole pitch is C and it connects an A minor triad to a D major-minor seventh chord. This is one of the uncommon instances in which the first chord is not a major triad. In the two statements of this musical idea, Mozart uses an FS and a TS.

Example 3-14a, K. 279/i, mm. 15-20

As expected, the main theme of the recapitulation also ends with a G major triad, shown in Example 3-14b. However, this time, the subordinate theme does not begin with D, the fifth of the chord. Rather, the second theme starts with G, the root of the chord. As in the exposition, the single note becomes the seventh of a major-minor seventh chord. The sequential pattern returns and the pitch F transforms from the third of a D minor triad into the seventh of a G major-minor seventh chord, which is of course, conveniently the dominant seventh of C major. Also following the pattern in the exposition, the first chord in the second two-measure idea is one of the rare (for common-tone transfers) minor triads.

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Example 3-14b, K. 279/i, mm. 68-73

To summarize, the subordinate theme in the exposition used an FS and the recapitulation uses an RS. This simple, subtle variation is typical of Mozart’s music. A single pitch connects the exposition to the development in this movement. The exposition ends in G major, but on the sole pitch G. The development immediately starts with a G minor triad in bar 39, startling us with the parallel minor. This is an RR. Unlike the subordinate theme, which began with a single pitch, the exposition ends on a single pitch.

Example 3-15, K. 279/i, mm. 37-39

In the second movement of K. 279, which is in F major, a single tone connects the main and secondary themes. In Example 3-16a, the exposition’s main theme concludes with a C major triad and the unaccompanied note G links to a G major-minor seventh chord that starts the subordinate theme at bar 11. This is an instance of one of the uncommon FRs I mentioned earlier. Perhaps Mozart found it more acceptable here because the fifth becomes the root of a dominant seventh chord, rather than a triad. The same idea occurs in the recapitulation, shown in Example 3-16b, but a C major triad now brings into play the pitch C (instead of G), which

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becomes the root of a C major-minor seventh chord (which again, is conveniently the dominant seventh of the home key).

Example 3-16a, K. 279/ii, mm. 9-12

Example 3-16b, K. 279/ii, mm. 49-52

In other words, the first theme of the exposition and the recapitulation both end with a C major triad, but in the exposition, Mozart uses an FR and in the recapitulation, Mozart uses an RR. *** I now turn my attention to the third movement of Mozart’s Sonata No. 4 (K. 282). The main theme begins with a single pitch, played twice. Shown in Example 3-17, the first two B-flats are harmonized with a tonic triad of the home key, E-flat major. The next two high B-flats are harmonized with B-flat major, resulting in a scarcely used FR. Mozart takes advantage of this apparently simple melody.

Example 3-17, K. 282/iii, mm. 1-3

Mozart bases the development on the main theme. In bar 40, found in Example 3-18, the development begins with two single B-flats, just as at

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the start of the piece. However, in the development, Mozart uses an RR in order to get us from B-flat major to B-flat minor. High B-flat is struck two more times, but now the B-flat becomes the seventh of a C major-minor seventh chord, which produces an RS. As the music continues to modulate, the repeated tone becomes A-flat instead of B-flat. In bars 43-45, Mozart harmonizes the A-flat with an F minor triad, an A-flat minor triad, and a B-flat major-minor seventh chord. As a result, we have two different common-tone transfers in a row, bringing about a TRS. The development becomes even more extreme with its use of commontone transfers in bars 48-54. The first E-flat is the root of an E-flat major triad; the second E-flat is the root of an E-flat major-minor seventh chord; the third E-flat is the fifth of an A-flat major triad; the fourth E-flat is the root of an E-flat minor triad; and the fifth E-flat is the seventh of an F major-minor seventh chord. Starting at bar 51, the next series of notes in the bass are Fs. The first F in bar 51 is the root of an F major triad; the second F is the root of an F major-minor seventh chord; the third F is the fifth of a B-flat minor triad; the fourth F is the root of an F minor triad; and the last F is the seventh of a G major-minor seventh chord.

Example 3-18, K. 282/iii, mm. 38-55

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mm. 47-51 Eb Eb Æ Eb7 Æ Ab Æ eb Æ F7 R R F R S mm. 51-55 F F Æ F7 Æ bb Æ f Æ G7 R R F R S Below Example 3-18, I have summarized the common-tone transfers in bars 47-55. In this movement, Mozart commences with a simple FR. By the development, we see how Mozart capitalizes on this understated idea. Mozart utilizes common-tone transfers in a multitude of ways in his piano sonatas. Although this paper barely reveals the tip of the iceberg, I believe that understanding these common-tone transfers will give us greater insight into Mozart’s compositions, as well as the evolution of the common tone.

Notes 1

Common-tone transfers stem from the theories of Moritz Hauptmann, who in his 1853 treatise Die Natur der Harmonik und der Metrik (cited hereafter as NHM), described the changed meaning of notes and chords. Trans. and ed. W.E. Heathcote as The Nature of Harmony and Metre, London 1888. All citings will include Hauptmann’s pages in brackets. 2 Hauptmann justifies this phenomenon by explaining that it takes no energy for the tonic to become the dominant of the subdominant. He writes when forming sevenths, going to the subdominant direction sounds “right,” whereas movement to the dominant direction sounds “fifthy or disjointed; in a word, wrong.” NHM, pp. 78-79 [par. 153, p. 103]. 3 This will be encountered later in the analysis of Mozart’s First Piano Sonata. 4 Interestingly, this is the same progression Brahms uses in the opening of his Third Symphony, which is referred to as a common-tone diminished seventh chord. 5 Curiously, when Hauptmann interprets these types of modulations—which he calls progressive modulations—he explains that it is the single note—not the chord—that has changed meaning. He writes: “…a relationship may also be contained in making the Root or Fifth of the tonic triad the Third, or its Third the Root or Fifth, of a new tonic triad.” NHM, p. 152 [par. 272, p. 181]. 6 According to Hauptmann, this type of modulation, in which third meaning transforms into fifth meaning, is to be avoided, most likely because he bases his reasoning on just intonation.

CHAPTER FOUR ANALYZING WAGNER’S “DER ENGEL”: QUESTIONS POSED AFTER APPLICATION OF RECENT TRANSFORMATIONAL THEORIES BARBORA GREGUSOVA

The Question and the Context What kind of results can be generated through detailed analysis of an entire work from the music of the New German School, the repertoire for which many recent transformational theories were originally designed? Currently, there are a limited number of analyses that examine larger sections or entire pieces. Detailed analysis of an entire work may reveal not only the strengths of the individual theories tested below, but also new areas for further development and future research. I analyzed Richard Wagner’s “Der Engel” from Wesendonck Lieder, written for high voice and piano, using the methodologies described by Richard Cohn in Audacious Euphony: Chromaticism and the Triad’s Second Nature and by Julian Hook in his “Uniform Triadic Transformations.”1 While I am aware of many other valuable transformational theories that could be applied (such as those by David Lewin, Robert Morris, Steven Rings, and Dmitri Tymoczko), my selfimposed limitation stems from the original intent of a detailed analysis of an entire work; presenting six individual analyses would take a monograph.2 Consequently, I selected methodologies that serve as representatives of contrasting monist and dualist approaches with regards to their targeted audiences. Cohn deemphasizes mirror symmetry and creates a “user-friendly” tool for analysis, while Hook’s dualist method is aimed at the professional music theorist. Consequently, I anticipate different results from these approaches. “Der Engel” is the first of five songs, all written in 1857-58 during Wagner’s stay at Otto and Mathilde Wesendonck’s estate in Zürich.

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Although Otto proved to be one of Wagner’s most valued patrons, Wagner put his support at risk when he engaged in an intimate relationship with Mathilde, a German poet, who supplied Wagner with artistic inspiration. He not only set five songs to Mathilde’s poems but also used two of them (nos. 3 and 5) as composition studies for Tristan und Isolde, an opera partially inspired by their romantic relationship. In 1858, Wagner was forced to leave the Wesendonck residence, or “Asyl,” which also meant the end of his relationship with Mathilde.3 Wesendonck Lieder was not published until 1900; however, the song cycle received several revisions and arrangements before the beginning of the twentieth century. Under Wagner’s supervision, it was first orchestrated by Felix Mottl, arranged for violin and piano in 1872 by Léonard, and later arranged for high voice and chamber orchestra by Hans Werner Henze in 1979.4 Furthermore, Wagner made an arrangement of “Träume,” the last song of the cycle, for violin and orchestra.5 In analyzing Wagner’s “Der Engel,” I start by making general comments regarding the piece’s organization and its formal structure, and then provide analyses according to Cohn and Hook. At the end of each section, I make remarks regarding any alterations created in order to account for aspects or elements omitted from discussion in these theories. Finally, I summarize the issues that arise during each analysis and point out some aspects of the composition not accounted for by the given theories. I focus specifically on identifying necessary alterations or bridging principles that suggest future paths for refinement of transformational theories that may assist with future analyses.6

Formal Organization The piece is organized in a traditional ternary form. After a brief introduction, the vocalist enters with a pick-up to m. 3. The A section spans mm. 1-13, including a brief introduction and two phrases, a1 (mm. 4-8) and a2 (mm. 9-13), respectively. Since few traditionally-conceived tonal progressions occur, the boundaries of individual phrases are determined by lyrical and textual correlation with the melodic contour. Similarly, changes in texture, text, general mood, and the number (or density) of voices divide the form into individual sections. The first instance of a minor chord (G-minor) on the downbeat of m. 14 signals the arrival of the B section.7 This consists of three semi-phrases: b1 (mm. 14-16), b2 (mm. 17-19) and b3 (mm. 20-23).8 The first two semiphrases can be considered parallel, since the vocal line of b2 is a real transposition of b1 up two semitones. The accompaniment, however, does

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not maintain the same (transposed) harmonic pattern. Measures 20-23, b3, could be either heard as the last phrase of the B section or, alternatively, as transitional material from B to A’, since the note values of the vocal line are augmented to resemble some rhythmic patterns from section A (such as m. 7 and m. 22). The return of the A material in the piano on the downbeat of m. 23 is followed in the voice on the next beat. Thus, retrospectively, mm. 20-23 can be relabeled as a part of the B section that is followed by the real transition in m. 23, extending through m. 28. As discussed previously, the A’ section arrives in m. 29 and consists of two phrases, a3 (mm. 29-32) and a4 (mm. 33-40). The last phrase overlaps with a postlude/coda and resembles the manner of articulation of the end of the transition and the beginning of the A’. Starting in m. 40, the last phrase is organized in three two-bar units. From the global perspective, the formal structure of the piece is ternary due to the dramatic change in texture of the accompaniment, the rate of change of pitches in the vocal line, and the overall switch to a new harmonic region (with prevalent minor sonorities and seventh chords of various qualities), which is followed by an imaginatively varied return. The structure of the recapitulation is continuous due to a lack of tonally orienting cadences and—while the A’ design material returns during the transition in mm. 23 or 24 (discussed later in greater detail)—it stays far removed from the original area of G-major. The opening harmony is delayed to m. 25 and preceded by its dominant sonority, which consequently grounds this familiar region through the use of a traditional tonal harmonic move. Furthermore, when the opening material finally appears in mm. 28-29 (accompaniment and melodic design respectively), Wagner opts for a false reprise by placing the return harmonically one semitone below the central region of G-major. The B section therefore concludes on a cadence outside of the original region and the reprise does not return in the home key; it only maintains the design. The second major interior cadence occurs in m. 13 at the boundary of the A and B sections. This point may be a bit more ambiguous since the vocal part and the accompaniment do not conclude the phrase simultaneously. While the singer closes on pc 7, offering connotations of home and suggesting a sectional character, the piano accompaniment moves from a G-major chord in the first inversion (the preceding D3 in the bass serves as an anticipation of the following sonority) and rests on a Ddominant seventh chord. In terms of tonal interpretation, this move concludes on a cadential formula with an evaded cadence and, as a result, forces one to rethink the meaning of the first D3 of the measure.9 Consequently, depending on the reader’s interpretation of the cadence in

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m. 13, the piece under examination is organized in either a sectional ternary or a continuous ternary form. In this paper, I treat “Der Engel” as a continuous ternary form and employ this formal analysis as a means of subdividing the harmonic language and examining the relationships between individual sections.

Analysis Analysis Using Cohn’s Methodology For the purpose of this essay, I limit Cohn’s theory to transformations of major and minor triads (diminished and augmented triadic sonorities do not occur in the given repertoire selection), as well as his reduction of dissonant chord members, chordal sevenths in dominants and roots of halfdiminished seventh chords into major or minor triads. All the aforementioned sonorities and their respective transformations are modeled via Tonnetze, each one based on a formal section of the work.10 Furthermore, Jack Douthett’s and Peter Steinbach’s “4-Cube Trio,” as presented in Cohn’s theory, is used to explain consecutive seventh chords; namely, the dominant, half-diminished, French augmented sixth chords and minor seventh chords.11 Lastly, I follow Cohn’s treatment of dissonance as an aid to uncovering the underlying harmonic progression. The analysis of “Der Engel” is divided into four Tonnetze, one for each formal section: A, B, Tr and A’.12 The moves between individual chords are displayed in two ways: visually, in the form of triangles (created by pcs as their vertices) moving through space, and as a linear sequence where each move includes a description of how one triad is transformed into the next.13 The dashed line, running vertically through the sequence, represents the boundary between two consecutive phrases or segments (for clarity’s sake, individual phrases are labeled below the chord sequence). Shown above the sequence of trichordal moves are measure numbers locating these sonorities. The A and A’ sections reside on the Tonnetz, as they do not feature consecutive seventh chords. For the B section, I present the trichordal version and also trace the seventh-chord succession on Jack Douthett’s “4-Cube Trio” as presented in Cohn.14 A Section Globally speaking, the first phrase of the A section begins on G major and moves West through a diatonic parallelogram, switching to another parallelogram at the beginning of the second phrase, as illustrated in

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Example 4-1.15 Finally, a neighborhood around pc 2 is used as a transition to the B section. Consequently, the overall “central neighborhood” of the A section switches from pc 7 to pc 2 (mm. 13-14) via pc 4 (mm. 9-11).16

Example 4-1, Cohn’s methodology applied to mm. 1-13 in a linear sequence and Tonnetz graph of section A

The means through which F major moves to E major at the beginning of the second phrase are described in my Conclusions section (bullet two under the Dissonance subsection on p. 12) and will receive a detailed discussion later in the essay. The second phrase is also interesting in that it does not remain in the confines of a diatonic parallelogram. In m. 9, E major moves down the octatonic diagonal, momentarily landing on a locally spelled-out triad with a root on C, and returning to E before ultimately moving on to A major. In this way, the move strengthens the pc 4 neighborhood by supplying an A-major triad to complete the chordal root, third and fifth within the neighborhood. The harmonic motion then progresses in a familiar direction of the diatonic parallelogram, giving an illusion of forgetting the D major and landing straight on G major. After a

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short detour back to the pc 4 neighborhood, the G major represents the beginning of another neighborhood around pc 2: G-D7-G-D7-B-flat-D7-g, turning to G minor at the beginning of the B section.17 B Section Section B (as shown in Example 4-2), in contrast to section A, is not structured around moves on diatonic parallelograms, but instead each phrase is centered round a pc neighborhood (pc t, 9, and 5, respectively). In the case of the second and the third phrases, these are located on a diagonal hexatonic axis created by a fully-diminished seventh chord. Furthermore, each phrase is removed by one hexatonic axis to the West from the previous phrase.

Example 4-2, Cohn’s methodology applied to mm. 14-23 in a linear sequence and Tonnetz graph of section B

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The first phrase starts out on a G-minor triad, which is transformed into a minor-minor seventh chord in m. 15 beat 3 and consequently allows for two interpretations: G minor and B-flat major. A quick move to E-flat major (the fourth beat of m. 15) before returning back to G minor (via the E-half-diminished seventh chord, m. 16) solidifies the neighborhood of pc t. G minor then moves over to far-removed A major, signaling the next phrase.18 Across the bar line of m. 17, the A-major chord moves to an F-major seventh chord, which covers two additional portions of the pc 9 neighborhood: F major or A minor, adding the D-minor chord on b. 4 of that measure. This region is maintained through moves between D minor (a subset of a B-half diminished seventh chord), A minor and F major sonorities; it does not alter until the arrival of a D-sharp fully-diminished seventh chord (m. 19 beat 2), which defines the octatonic axis on the Tonnetz about which A minor (F-sharp half-diminished seventh chord) moves to B major (second half of m. 19).19 Marking the beginning of the third phrase (m. 19), B major moves to its Nebenverwandt, E minor, before moving down the hexatonic diagonal to B-flat major (m. 21). B-flat major, along with D minor (B-halfdiminished seventh in m. 22) and F major, create a neighborhood around pc 5. D minor then moves to E major (m. 23) via an axis created by the Gsharp fully-diminished seventh chord (m. 22 b. 4), removed by one to the left from the previous axis of D sharp.20 Therefore, the ends of phrases 2 and 3 are approached through the same transformation, but transposed to the left by LR on a diatonic parallelogram. Transition In the transition between B and A’, shown in Example 4-3, E major (m. 23) marks the beginning of the return of the design, but not the beginning of the A’ section. The sonority is immediately transformed into an augmented sixth chord, which is impossible to draw out on the Tonnetz, but Douthett/Steinbach’s “4 Cube-Trio” shows this move through the network in the following section. After the initial far-removed leap to D major (m. 24), the harmony moves back and forth on a diatonic parallelogram, emphasizing the pc 2 neighborhood. Ultimately, it moves West on the parallelogram to E minor (or a C-sharp half-diminished seventh chord). This serves as a passing point and marks both the end of the transition and the beginning of the first phrase of the reprise, which starts with another far-removed sonority: an F-sharp major triad in m. 28.21

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Example 4-3, Cohn’s methodology applied to mm. 23-28 in a linear sequence and Tonnetz graph of section Tr

A’ Section The linear sequence for Section A’ and its associated Tonnetz are given in Example 4-4. Measure 28 references a temporally diminuted version of the introduction of the entire piece, but at the wrong pitch level. In fact, in global terms, it follows the plan of the A section in that it also moves West on the diatonic parallelogram, hitting the pc 4 neighborhood as a way of getting to pc 2.

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Example 4-4, Cohn’s methodology applied to mm. 28-45 in a linear sequence, and Tonnetz graph of section A’

The false reprise begins down by a semitone, text-painting the angel that “dips down” to bear the soul to the heavens above. F-sharp major transforms to E major, arriving in m. 30, which together with C-sharp minor (m. 29), A major (m. 31) and C major (m. 33-34) create the neighborhood of pc 4.22 The return back to the neighborhood of pc 2 begins in m. 38. G major moves to a B-minor seventh chord, which can be interpreted as either B minor or D major (both a part of the same neighborhood), before arriving on the final G major. It is then sustained by one more move, located West on the parallelogram (C major) as a part of the coda until it returns to the last G major chord. Douthett/Steinbach Cube23 Example 4-5 presents part of the song plotted on Douthett/Steinbach’s 4-Cube Trio, as altered by Cohn. The analysis is limited to mm. 16-29 because the inclusion of the other parts of the piece would require constant switching from this model to the Tonnetz, and they have already been covered above. On the diagram, each sonority is labeled with its exact

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Example 4-5, Cohn’s 4-Cube Trio (from Douthett/Steinbach’s theory), mm. 16-29 (section B+Tr)

location within the piece, as well as an ordinal number for the sake of clarity.24 The directed dashed line represents the moves through the 4Cube space. Furthermore, sonorities marked with an “X” through their vertices represent the ones that appear in the score. Consequently, if the dashed trajectory moves through a sonority not marked with an X, it indicates that a gap in parsimonious voice leading occurs in that place. Such instances occur on chords on 4, 8, and 10, with 4 and 8 creating

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mirror symmetry around vertex 0, the only augmented sonority used in “Der Engel.” One can observe that the sonorities used in the majority of the B section can be nearly seamlessly traced counter-clockwise all the way around the 4-Cube Trio, ending two moves past the point of origin. With such a high rate of change in individual sonorities, and pattern-like, predictable motion around a cyclic space, possible relations to tonal sequences can be witnessed. Could Wagner have been considering the connotations of the B section as development? This question must necessarily remain unanswered here. Conclusions From the Analysis Cohn’s approach is successful in clearly modeling the space in which the piece moves. Therefore, any lingering within the same area may be thought of as centricity; on the other hand, disjunct motion can be visually detected on Cohn’s Tonnetz and, consequently, easily addressed from the global/local standpoints of importance for the entire piece.25 An example of such motion is located at the end of phrases within the vocal line in “Der Engel,” which features a sonority far removed from the previous moves on the Tonnetz. This suggests that the end of the vocal line is actually the beginning of the next phrase, started in the accompaniment, and is designed to clearly delineate the pc areas of each phrase. Consequently, this technique nicely juxtaposes and dovetails the voice and the accompaniment. The two possible avenues for further development of Cohn’s theory are: (1) more detailed definitions of all dissonance types, and (2) ideas about generation and treatment of some omitted seventh chords.26 Following are some examples that arose while applying Cohn’s theory in my analysis. Dissonance 1. Following the principle that a slower moving harmonic rhythm in the accompaniment gives rise to the distinction between chordal members and dissonances in the vocal part, the move from the D (downbeat of m. 4) to the G (downbeat of m. 5) could be explained as a double passing tone between these two pitches—a phenomenon that neither Cohn nor Hook have accounted for in their theories. While this case is easily solved with standard contrapuntal readings, in cases where the texture does not offer a simple and unambiguous harmony, one would benefit from hard or soft

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rules for interpretation. Consequently, although the lack of set rules for dealing with metric/agogic issues allows for the analyst’s interpretation based on the context and era of the piece, bridging principles would be helpful as a guideline for dealing with more complex temporal issues. Since these bridging principles have not been formulated, both options are presented in my Tonnetz figures, including the chord-to-chord hierarchyignoring version.27 The same situation occurs in m. 5. 2. The pick-up to the next phrase (m. 8 beat 4) presents an R-move from F major to D minor. The D-minor chord is placed on a weaker part of the measure and removes itself farther away from the consecutive chord, E major. This would suggest the treatment of D minor as a contrapuntal means, similar to the E-minor chord used as neighborhood-related chord to G major in mm. 4 and 5. However, dissonance treatment rules have not been clearly defined. Therefore, both options are included in the analysis.28 3. The G minor harmony in m. 14 beat 4 features an A and a C appearing above the harmony. As these pitches do not create any type of a trichord (or even a seventh chord) they are treated as metrically-shifted and durationally-distorted ornamental tones. The C5 is a clear upper neighbor to Bb4, while the A4 makes for a less possible double neighbor figure due to the slight agogic and metric distortions. The treatment of neighbors needs to be discussed as a part of dissonance options. 4. The C-sharp minor chord locally formed on beat 3 of m. 29 is subordinate to F-sharp major when considering the metric/agogic accent of individual sonorities, as discussed in previous points. Nevertheless, the subordinate C-sharp minor sonority participates in the completion of the pc 4 neighborhood. Major and Minor Seventh Chords 1. In m. 11 beat 3, the E minor seventh chord poses another question for bridging principles. It contains two possible triads, E minor and G major. While both options are always provided in my analysis, I prefer the E minor because—in the context of this piece—Wagner always places the minor seventh chord next to its subset (G major can be created out of the E minor seventh chord). Therefore, the chord that generates a move, rather than an expansion, will always be preferred throughout the analysis. The second possibility for interpretation would be explaining the D in the accompaniment of m. 11, beat 3 as a suspension, which resolves down to C# (the chordal third of the immediately following A dominant seventhchord). Furthermore, following the weak metric placement/fast-moving sonorities as a means of hierarchical structure, A7 can be reduced as a

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lower neighbor chord to its surrounding sonorities. Eliminating the A7, as well as the chordal seventh of the E minor seventh chord, would result in a G-major expansion of G-e-G, identical to the one presented in m. 4. The interpolated A7 is, therefore, subordinated to E minor (seventh chord), which is then inferior to G major by participating in its expansion.29 2. Measure 15, beat 3 can also be interpreted using m. 4 as a model, resulting in a simplified analysis. If the move from F to E in the vocal part is considered as double passing tones—moving down from G on beat 2 to Eb on beat 4—both the G minor seventh chord and the E half diminished seventh-chord would be eliminated, thereby providing for a simple move from G minor to E-flat major. The E-flat sonority then serves as a neighborhood expansion of G minor (similarly to the treatment of mm. 45); the E♭ major is subordinate to G-minor on metric/agogic grounds. The G minor sonority arrives again on the downbeat of m. 16 with its underthird as a dissonance (E half diminished seventh chord). As modeled in my analysis of “Der Engel,” major and minor seventh chords acquire the ability to function as two related triads and provide a means of identifying neighborhoods. From my limited sample size on which the idea was tested thus far, it appears that these seventh-chord sonorities are used as an expansion of a minor or major triad by adding an under-third (e.g. the E minor seventh chord is created by adding a third to a G major chord, such as in m. 11, beat 3).30 The advantage to using such seventh chords is the acquisition of two possible functions (a “doubleagent,” as discussed in Cohn, 68-78): more options for transformational moves and the ability to cover more neighborhood ground at once. All of the above could be used as valuable analytical tools; however, this idea is currently a limited hypothesis that requires further careful observation and additional score examination for potential uses. Further Questions The change in the nature of the accompaniment of the B section as well as the difference in harmonic rhythm, suggested by the modal mixture, pose more methodological questions that have not been addressed by Cohn or by my current analysis. These include, but are not limited to the following: (1) the issue of the “outlier,” (2) the importance of the root/inversions, and (3) a possible issue of more developed hierarchies.31

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Analysis Using Hook’s Methodology For the purposes of this essay, I have limited my use of Hook’s Uniform Triadic Transformations to his inclusion relations of major and minor triads using one of the transitive subgroups (R), as well as seventh chords accounted for in his theory (dominant and half-diminished seventh chords).32 Since Hook’s theory is limited to the transformations of major and minor triads (and seventh chords), and is in a certain sense even more limited than Cohn’s theory in that it accounts for fewer triads and seventh chord qualities, I used the segments developed for my Cohn analysis as a basis for using Hook’s methods (presented in Examples 4-6 through 4-9) and determined which transformations are most commonly used as ways of moving within the individual sections of the piece, as tabulated in Example 4-10.33

Example 4-6, Example of application of Hook’s UTTs on section A (mm. 1-13)

Example 4-7, Example of application of Hook’s UTTs on section B (mm. 14-23)

Example 4-8, Example of application of Hook’s UTTs on section Tr (mm. 23-28)

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Example 4-9, Example of application of Hook’s UTTs on section A’ (mm. 28-45)

Example 4-10, Table overview of UTTs in Wagner’s “Der Engel”

The table in Example 4-10 shows all UTTs that appear in the piece more than once (or at least once in the case of transformations closely related to other transformations with high frequency of use). In order to show relationships between individual transformations and sections, these are first presented per section and, finally, through a cumulative sum of appearances. The individual transformations are presented in both their UTT form and their Riemannian Sn/Wn form located in the third and second column from the left, respectively. The first column represents an overview of larger sections of the piece and their respective typical transformations. Transformation appears both in A and A’, but undergoes a transformation in the B section, where the intervals of transposition remain the same while the mode changes. The A section also includes a modepreserving third-relation transformation , as well as its inverse.34 The B section also introduces a related transformation not yet used in the entire piece, /, transposing the entities by a major third or minor sixth while changing the mode.

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The A section’s moves between individual chords are also generated through , a mode-preserving move by a perfect fifth on Cohn’s diatonic parallelogram. This transformation repeats in the A’; however, the reprise also introduces movement in the opposite direction on the parallelogram via . Moreover, the A section features transposition by the same interval while changing the mode () and its inverse () in the A’ section. The transformation and its inverse also appear in the B section as well as in the transition. Lastly, is also used consistently throughout the piece and maintains associativity while providing a comfortable familiar thread (in a psychological sense) while other transformations change. Example 4-11 presents the chord successions with the segmentation found in Examples 4-6 through 4-9 using the Riemannian Transformation Rsubgroup labeling.35 In the Sn/Wn notation, S (schritt) and W (wechsel)—from Riemann’s terminology and later adapted by Henry Klumpenhouwer, Hook and others—stand for mode-preserving and mode-reversing transformations, respectively. The subscript, n, represents the number of semitones by which a major triad is transposed, while a minor triad is transposed up by –n (which is equivalent to a downward transposition by n).36 The first set of successions, representing section A of the piece, presents all linear sequence transformations as mode-preserving moves, generally moving by five or ten semitones (in the case of A major to G major). One may think of this distant move as two sets of S5 transformations with an omitted statement of the D-major harmony. This may serve as a text painting of the distance that one needs to travel in order to “exchange Heaven’s sublime bliss for the Earth’s sun.” The farremoved G major is then followed by an S7 transformation to D major, which is followed closely by an S5 move back to G major. Following the principles of combining multiple transformations presented in Hook’s theory, where SmSn=Sm+n, the transformation G-D-G represents So, or identity, and can therefore be thought of in terms of sustaining G major.37 Similarly, all other possible successions considered in Cohn analysis, labeled as vertical pathways diverging from the linear sequence, add up to S0. Consequently, since neither the alternative paths nor the triadsustaining sonorities change the global plan of structurally-important transformations, these instances will be eliminated from the discussion. The other two exceptions from the five-semitone transformation plan include (another mode-preserving) S11 between F major and E major, whose meaning is explored later in the essay, and the mode-reversing W5, which marks the move to the contrasting B section.

Example 4-11, Hook’s Riemannian R-subgroup notation

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Comparing section A with its A’ return, the moves between chords nos. 2-8 in A and nos. 5-11 in A’ are identical, with the exception of the chord nos. 7 and 10, whose consideration for discussion is eliminated due to reasons stated above.38 This sequence is then followed by a coda, which maintains G major.39 The first four chords preceding the chord succession from the opening of the piece follow the five and ten semitone transformations from the original A section. The only exception is present in the move into the seven-entity chord succession, the S8 transformation from E major to C major, which marks the first move back into the original region of G major or “back to Heaven.” Before C major, the false reprise in F-sharp major seemed to be distancing itself from Heaven by heading deeper into the lower register in the accompaniment until m. 33, the point at which the angel reaches the soul and starts to head towards Heaven. The B+Tr section consists primarily of five- and ten-semitone transformations as well; however, the contrast stems not only from the change in the texture and the move to a minor mode region—it consists mainly of mode-reversing transformations, which may also suggest the contrast between Heaven and Earth. The move from A major to F major to D minor uses a cumulative transformation of W5 (in terms of Hook’s principles of addition of multiple transformations), following, less clearly in this instance, the large scale plan of chordal succession. Another less common move occurs on chord no. 7, where E minor moves to B-flat major to D minor. These transformations are reduced to S2, which seemingly does not correspond with other surrounding transformations; however, the S2 outlines a move between two minor triads that are removed by ten semitones—a distance recurrent in the structure. Finally, the only transformation falling outside of the presented theory occurs at the end of the middle section, leading from G major to F-sharp major (the false reprise) through S11. This leads me to address the importance of this transformation. It appears exactly once in each section, moving from F major to E major in both A and A’ and from G major to Fsharp major as a move from the B section back to the A material; each time it marks the important move from one place to the next—whether from the Earth to Heaven or from Heaven to Earth—and emphasizes the distance between the two worlds. This idea is potentially represented through semitonally-related triads having maximally-contrasting pitchclass sets, the moves between which are used as text painting within the pitch structure.

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Inclusion Transformations With Seventh Chords Example 4-12 includes mm. 14-23, the B section, which almost exactly corresponds with the passage presented in Example 4-5, Douthett/ Steinbach’s 4-Cube Trio. As per Hook’s model, the sonorities in the upper line represent major and minor triads, while the lower line includes seventh chords. Since the majority of this essay focuses on triadic transformations, they are omitted from Example 4-12; included are only transformations between triads and seventh chords and between two consecutive seventh chords. The transformations of inclusion between triads and seventh-chords are labeled with “⊂” where a triad becomes a subset of a seventh-chord, while “⊃” represents its inverse.40 Measures 1718 take advantage of the inclusion relation and lengthen the F major sonority by the use of a retrograde of a three-chord sequence F-d-bØ7 with the B-half-diminished seventh-chord acting as its axis.

Example 4-12, Hook’s UTT seventh-chord transformations (mm. 14-23)

When tracing the transformations between the individual halfdiminished seventh chords, they are first transposed up by a perfect fifth with roots E-B-F#, with the last sonority acting as an axis of retrograde back down to E.41 The last sonority with its root on E, however, differs from the E-half-diminished seventh chord by one pc and creates an augmented sixth chord that is used as a bridge leading up to the transition. The two moves of S5, prior to reaching the F# axis, do not simply emphasize the importance of five and ten semitones (S5S5=S10) and the identity of the entire section created by the retrograde of the two S5 transformations (arriving back to S0 on E); they also point to the relationship between the F# and E that is explicitly laid out in the false reprise. It is precisely at the moment of the F# axis that the lyrics switch

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from describing the despair on Earth to the soul’s preparation to plead for a release to Heaven—a small hint at what comes next—as the transformations take turn back to the root of E. Outlined with a box in Example 4-12 are chord successions with identical transformations of or W7 (cells), which represent the perfect fourth displacement of the root while changing quality of the seventh chord from half-diminished to dominant. While the first one of these instances is missing the fully-diminished seventh chord with which the transformation occurs (for visual representation, see the Tonnetz in Ex. 4-2), the second two successions are related by T5 or S7—the same interval that the roots between the consecutive seventh chords are displaced by within the individual successions. One may question the relevance of the chord succession cells and their relationship for two reasons: (1) the first cell is missing the fullydiminished link between the two fourth-related seventh chords, and (2) the second half-diminished seventh chord of the sequence (the very same bØ7 about which the “prolongation” of F-major occurs) shows no signs of such a cell and is undermined in importance due to lack of accent (whether durational or metric). Both of these concerns can be explained using Hook’s discussion of inversionally symmetric sets.42 According to Hook, “there is no real need to apply UTTs to inversionally symmetric sets. Such a set has only a single mode, so the sign and the second transpositional level of a UTT acting on such a set are irrelevant.”43 Due to the nature of inversionally symmetric sets and their limited number of transformations, the UTT methodology would, in a narrow sense, model Tn/TnI. Consequently, since this essay models in detail one of Hook’s transitive groups, the Riemann group and not Tn/TnI, this issue is not within the confines of the discussion; therefore, less importance will be assigned to inversionally symmetric chords. Within the excerpt in question (mm. 14-23), there are four instances of inversionally symmetric sets. There are two instances of fully-diminished seventh chords in the third and fourth cell (mm. 19 and 22): an F-majormajor seventh chord in m. 17 and a D-minor-minor seventh chord directly preceding the B-half-diminished seventh in m. 18. Lessening the importance of the diminished seventh chords validates the first occurrence of the cell, which is presented without such an entity. Furthermore, eliminating the major-major and minor-minor seventh chords would, on one hand, refute the F-major “prolongation” discussed earlier but on the other hand it would permit the B-half-diminished seventh chord to gain importance by allowing it to acquire both durational and metric accents. While this still does not account for the lack of a cell in m. 18, it provides

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a possible explanation for this chord as a structurally important point within the B section.44 Hook’s methodology for transformations on seventh chords uncovered additional relationships and symmetries that were not exposed using the 4Cube Trio. Consequently, it supported further the formal function of the B section’s development/digression. Conclusions From the Analysis While the success of Hook’s methodology may be very dependent upon the piece to which it is subjected, the theory has the potential to give some insight into the more global organization of a piece.45 Although some of these transformations could be spotted at first glance (e.g. the implementation of moves by a perfect fifth/fourth, major and minor third), this theory presents the information in a more comprehensive way, providing the analysis with clarity and rigor. Furthermore, Hook’s theory has the ability to provide plausible information regarding the problem of the “outlier” and its relationship to the global form, as well as to detect mirror symmetries thanks to its use of Riemannian dualist ideas, which may prove to provide additional information about the piece’s large-scale structure.

Conclusions and Needed Bridging Principles The goal of this paper was to consider Cohn’s and Hook’s theories for analysis using a piece from the repertoire for which they were originally designed. While both theories helped to shed some light on the local and, to some extent, global organization of Wagner’s “Der Engel,” the following issues (either omitted or not addressed in the theories) require bridging principles in order to account for all aspects of the composition.

Issues Not Addressed in Either of the Methodologies46 1. Cadence/closure issue: In harmonic scale-step theory, the idea of closure is strongly related to the harmonic progression, the linking of contrapuntal patterns, as well as concluding typical phrase lengths (slight variations are possible). Its absence in the repertoire of the mid-nineteenth through early twentieth century suggests that a new definition of closure for Transformational theories, applicable specifically to Neo-Riemannian theory, needs to be

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2.

3. 4. 5.

created.47 For pieces not concerned with closure, this issue does not need to be addressed. Omitted qualities of chords: Further discussion is needed on the issues of commonly found trichords and tetrachords in this repertoire. Chord qualities not explored in Cohn’s theory include: diminished and (to some extent) augmented triads, major and minor seventh chords and fully-diminished seventh chords, as well as other non-traditional chord types (stacked fourths, fifths, etc.). The “outlier” complex: The issue and function of a chord far removed from the rest of the related moves needs to be addressed. Metrical ambiguity: Metrical ambiguity (with potential to create local dissonances) due to placement. Implied pitches: Lack of information may result in several possible paths. Bridging principles are needed in order to determine which path to choose.

Issues Addressed in the Given Theories, But That Need to be Revisited Due to Problems That Arose During Analysis Issues with the definition of a passing seventh include the following: 1. Instances when a passing seventh does not resolve in the traditional way down by a step, 2. According to Cohn’s theory, the passing seventh is omitted from the analysis in order to preserve a major/minor triad. In the case of a passing seventh in an outer voice, bridging principles need to be designed in order to account for its other possible function,48 3. The question of a passing seventh exposes a temporal issue thus far unexplored in transformational theory. The above points are only the ones that pertain to my analysis of “Der Engel.” I suspect that while some (if not all) of these questions shall remain, other issues might arise when examining different works. Consequently, my analysis of “Der Engel,” using Cohn’s Audacious Euphony and Hook’s UTT principles, is a preliminary study designed to point out some of the issues not addressed by the theories at hand, which will ultimately lead to further beneficial development of Neo-Riemannian or Transformational Theory.

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Notes 1

See Richard Wagner, “Der Engel,” Wesendonck-Lieder (New York: C.F. Peters, 1900); Richard Cohn, Audacious Euphony: Chromaticism and the Triad’s Second Nature (Oxford: Oxford University Press, 2012); Julian Hook, “Uniform Triadic Transformations,” in Journal of Music Theory 46, nos. 1/2 (2002): 57-126. 2 David Lewin, Generalized Musical Intervals and Transformations (Oxford: Oxford University Press, 2006); Robert D. Morris, “Voice-Leading Spaces,” in Music Theory Spectrum 20, no. 2 (1998): 175-208; Steven Rings, Tonality and Transformations (Oxford: Oxford University Press, 2011); Dimitri Tymoczko, A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice (Oxford: Oxford University Press, 2011). 3 German for Asylum. Barry Millington, “Wesendonck, Mathilde,” in Grove Music Online, Oxford Music Online, accessed June 6, 2013, http://www.oxfordmusiconline.com/subscriber/article/grove/music/30144. 4 Also see The Oxford Dictionary of Music, “Wesendonck Lieder, 5 (Fünf Gedichte von Mathilde Wesendonck),” in Oxford Music Online, accessed June 6, 2013, http://www.oxfordmusiconline.com/subscriber/article/opr/t237/e10996?q= Wesendonk+Lieder&search=quick&pos=4&_start=1#firsthit. 5 My analysis only discusses the issues raised in “Der Engel.” For analytical work on “Träume” see Robert Gauldin, “Wagner’s Parody Technique: ‘Träume’ and the ‘Tristan’ Love Duet,” in Music Theory Spectrum 1 (1979): 35-42. 6 Throughout the essay, I will refer to these alterations as ‘bridging principles’. Frederick Suppe adapted Hempel’s proposition of two principles essential to a theory: 1) internal principles, “concerned with the peculiar entities and processes assumed by the theory, expressed largely in terms of characteristic ‘theoretical concepts’,” and 2) bridge principles, “connecting certain theoretically assumed entities that cannot be directly observed or measured with more or less observable systems.” Furthermore, Suppe suggests, “the phenomena to which bridge principles link the basic entities and processes assumed by a theory need not be ‘directly’ observable or measurable: they may well be characterized in terms of previously established theories, and their observations or measurement may presuppose the principles of those theories.” For complete definitions, see Carl G. Hempel, Philosophy of Natural Science (Englewood Cliffs: Prentice-Hall, 1966), pp. 73-74. Also see Frederick Suppe, The Structure of Scientific Theories (Urbana: University of Illinois Press, 1977). 7 Although several instances of minor chord sonorities occur previously (such as in mm. 4, 5, 8 and 11), all are questionable due to metrical placement and lack of agogic accent and therefore function as passing sonorities rather than structural ones. Conversely, the G-minor sonority spans nearly two full measures and— strengthened by the change in texture from arpeggiations to block chords— becomes aurally more hierarchically important. Consequently, the emphasis on the G-minor sonority reinforces the move to the B section by a shift to a modal mixture area (or neighborhood of pc t) on a global scale.

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Cohn’s methodology defines neighborhood as “pcx [where pcx represents any one pc] connected to the six tones with which it is consonant [creating six edges]. Six further edges, forming a hexagon, build those six tones directly to each other by consonances. The hexagons and radii together form six triangles, representing the six triads that include [pcx] (as root, third, or fifth of a major or minor triad).” See Cohn, 113. Therefore, a neighborhood consists of all major and minor triads that contain a single pc in common. 8 No cadential point occurs; however, the three individual semi-phrases can also be thought of as a sentence that exhibits features of a sentence structure, but with unusual proportions of 1:1:3 instead of the traditional 1:1:2. According to Caplin, “the contrasting middle of the small ternary is more loosely organized than the preceding exposition.... The B section is usually less symmetrical than the A section primarily by means of harmony and phrase structure.” See William E. Caplin, Classical Form: A Theory of Formal Functions for the Instrumental Music of Haydn, Mozart, and Beethoven (Oxford: Oxford University Press, 1998). 9 For the purposes of this essay, I use the pitch labeling system of the American Acoustical Society. 10 Refer to Examples 4-1 through 4-4 for individual Tonnetz representations. Moves between individual structural chords are labeled with an arrow and an ordinal number starting with “1.” at the beginning of the given section. All other chord-to-chord (non-structural) moves are labeled with a dashed arrow, as well as an ordinal number of the move identical to the next structural move. Non-structural moves will be also demarcated with an “*” next to the ordinal number. These numbers shall not be confused with measure numbers labeled at the top of the figure above the linear sequence. Circles around individual pcs represent neighborhoods or a collection of chords where all major and minor triads have a single (circled) pc in common. For a more thorough definition of neighborhoods, see footnote 7. Chords in an adjacent horizontal alignment form diatonic collections; chords in a diagonal top left to bottom right (NW to SE) form an octatonic (8-28 [0134679t]) collection, while chords in a diagonal top right to bottom left (NE to SW) form a hexatonic (6-20 [014589]) collection. 11 Jack Douthett and Peter Steinbach, “Parsimonious Graphs: A Study in Parsimony, Contextual Transformations, and Modes of Limited Trasposition,” in Journal of Music Theory 42, no. 2, Neo-Riemannian Theory (1998): 241-63. 12 The pcs Tonnetz, used in my examples, was designed by Prof. Richard Hermann in 2013. It accounts for equal temperament and furthermore, carries the advantage of displaying in a more clear fashion close relationships and transformations with sonorities where enharmonic notation is involved. 13 Located directly above the Tonnetz representation. 14 Cohn, 158. 15 An alternative reading for mm. 4-5 is presented in Conclusions From the Analysis under section Dissonance, 1 (p. 12). In accordance with this reading, the G-e-C chord succession creates a neighborhood of pc 7, before moving on to F-

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major towards the end of the phrase. For the moves on the Tonnetz, the map reading convention of the top being North, the bottom South, etc., is applied. 16 Ibid., 114. 17 An alternative reading and further notes about this move are included in the Conclusions section under “Major and minor seventh chords” (p. 12). 18 An alternative interpretation considers m. 16 as a double suspension of D4 to C#4 and Bb4 to A4, eliminating the half-diminished entity. 19 Similarly to mm. 4 and 15, a double passing tone appears on m. 18, beat 3. 20 See Ex. 4-2’s Tonnetz, where the axis is represented by a dashed line. Usually describing symmetrical inversion in p- or pc-space, for the purposes of this essay, axis represents the diagonal lines of symmetrical entities, as visually represented on a Tonnetz. Fully-diminished seventh chords are displayed from North-West to South-East (Hexatonic), while the augmented triads run North-East to South-West (Octatonic). 21 D major, G major and B minor create the neighborhood of pc 2. The B minor triad’s chordal fifth, however, comes from the temporally-suspended chordal third of D major in the previous measure and acts as a pedal point until the F-sharp major sonority in m. 28. The F-sharp major chord, along with D major and B minor (m. 24) create a second neighborhood around pc 6. Cohn’s methodology accounts for two simultaneous neighborhoods. Using Brahms’s Second Symphony, Cohn explains the move between neighborhoods of pc 2 and 9 as “a counterclockwise tour of the space, from subdominant to subdominant, with some back-filling that extends the retention loop around the dominant A and delays the premature return to the tonic region.” Further, he notes that the “pitches serve not as roots of tonic and dominant triads, but rather as thirds and fifths that disperse inventively across their respective neighborhoods.” See Cohn, 117-21. Since pc 2 and 6 are not related by a diatonic parallelogram and two sonorities are used as members of both neighborhoods, a different interpretation is needed. Pitch-class neighborhoods 2 and 6 (although farther removed) highlight the commonalities between the transition and the A’ section. Although the transition begins on an E dominant seventh, it is not until the D major chord in m. 24 that the “angel-like” accompaniment material (characteristic for A and A’ sections) appears. Is it a coincidence that the chordal roots of the return of the “angel” design in the accompaniment and in the false reprise (neither placed where expected; in terms of formal organization and harmonically, respectively) also represent the pc neighborhoods of these sections? For a more seamless transition between Tr and A’, both sections begin with a shift of the first sonority down by two semitones (E major to D major and F sharp major to E major). 22 See discussion in Conclusions under Dissonance, 4. 23 Cohn’s 4-Cube Trio differs from Douthett/Steinbach’s “Power Towers” in that the figure is rotated in space so that the vertex “0” starts in the north, while “Power Towers” begin in the southwest corner. See Douthett and Steinbach, 256. While the trajectories between individual sonorities remain the same, Cohn’s placement of these sonorities within the vertices are of opposite order. In addition, 4-Cube Trio accounts for augmented-sixth chords as possible routes of movement.

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The ordinal numbers of individual chords are circled in Ex. 4-5. Joseph Straus defines centricity as organization in music around referential pc centers or collections. “In the absence of functional harmony and traditional voice leading, composers use a variety of contextual means of reinforcement. In the most general sense, notes that are stated frequently, sustained at length, placed in a registral extreme, played loudly, and rhythmically or metrically stressed tend to have priority over notes that don’t have those attitudes.” See Joseph N. Straus, Introduction to Post-Tonal Theory (New Jersey: Pearson Education, 2005), p. 131. 26 Both are described in detail in Conclusions and Needed Bridging Principles. 27 All other possible options for moves are always displayed below the most direct move in the sequential graph. Furthermore, these hierarchical moments that locally spell out a sonority, but do not function as one in the global perspective, are marked in parenthesis in Ex. 4-1, or in dashed parenthesis if the sonority exhibits some degree of a structural function. On the Tonnetz in Ex. 4-5, they are labeled with a dashed arrow, as well as an ordinal number of the move identical to the next structural move. Non-structural moves are demarcated with an “*” next to the ordinal number. 28 Given the slow harmonic rhythm, the D-minor triad could also be heard as a consonant passing tone or a 5-6 suspension. 29 While this is a part of my generalization, bridging principles would be useful to account for minor seventh chords and their treatment when it is inefficient to invoke 4-cube analysis. 30 My application of this theory is currently limited to pieces by Wagner, Giuseppe Verdi and Zdeněk Fibich. 31 An example of these hierarchies could be demonstrated by chords far removed from the preceding and/or following sonorities (such as mm. 23-24), which—due to their frequency—appear to be of structural importance, but can not be explained using Cohn methodology. 32 According to Hook, Uniform Triadic Transformations (UTTs) represent triadic transformations formulated through simple algebraic framework using a rootinterval approach, which was designed to address problematic issues within various neo-Riemannian theories. See Hook, 59. Hook states, “the selection of one simply transitive group leads to a unique transformational analysis of any triadic progression, but different choices of the group may yield different (yet still meaningful) analyses.” See Hook, 60. While I am aware that various analyses using other transitive subgroups are possible and may lead to fruitful discoveries about the structure of the piece at hand, presenting detailed analyses using all possible subgroups is beyond the scope of this essay and will not be explored by me at this point in time. 33 Consequently, Hook’s theory would benefit from the same bridging principles as does Cohn’s theory. The UTTs consist of a sign (+ or –, indicating whether the transformation preserves or reverses mode of the triad) and two transposition levels (integers of mod12, which indicate the interval through which the root of the triad is transposed, major and minor respectively). For example, for given modereversing transformation: 25

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X= major root transposition in pc-space Y= minor root transposition in pc-space, both D major to G minor and G minor to D major are the products of the same transformation. Since these transformations are involutions as they are not concerned with the pc-space voice-leading, Hook suggests that they “may be regarded as explicit representation of Riemann’s well-known harmonic dualism.” See Hook, 59. While other, non-simply transitive transformations are presented by Hook’s theory, only Riemannian Transformations (transformations exhibiting this property) will be considered in this essay. 34 In this context, the term inverse is used to describe a transformation of the UTT that interchanges its transposition levels, as well as inverts them. According to Hook, “the inverse of a mode-preserving UTT is , while the inverse of a mode-reversing UTT is .” See Hook, 71. In these terms, an inverse of a transformation is . 35 A variation in segmentation is limited to sections B and Tr, which are presented in a combined form in Ex. 4-11. 36 Henry Klumpenhouwer, “Some Remarks on the Use of Riemann Transformations,” in Music Theory Online 0.9 (1994), accessed April 14, 2013, http://www.mtosmt.org/issues/mto.94.0.9/mto.94.0.9.klumpenhouwer.art. 37 The following formulas are derived from the product and inverse proofs presented by Hook: SmSn=Sm+n SmWn=Wm+n WmSn=Wm-n WmWn=Sm-n For proofs of the above formulas, see Hook, 75. 38 The numbers were assigned to chords sequentially based on their appearance in the piece, starting with no. 1 at the beginning of each individual sequence (A, B+Tr and A’). 39 If one were to view the coda in tonal terms, the plagal motion could be attributed to the sacred nature of the text. 40 “The inverse of ⊂, denoted ⊃, is the seventh-chord-to-triad transformation that maps each seventh chord to the unique major or minor triad that it contains.” See Hook, 117-18. 41 Transpositions (T5 and T7) are labeled S5 and S7, respectively, as per Hook’s terminology. 42 Although half-diminished seventh chords are not inversionally symmetric, this part of Hook’s theory sheds some light on the importance of this half-diminished sonority. 43 Ibid., 113. 44 This may be due to emphasis on S10, rather than two S5 transformations.

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45 For this study, my measure of success depends upon the statistics associated with counting the instances of particular transformations that occur within the individual sections (and globally) throughout the piece. This method, however, would not translate well to pieces that do not possess sufficient instances of cause and effect and/or transitivity. 46 While not all cases of dissonance treatment were accounted for by the theories, Wagner’s “Der Engel” does not feature any cases of dissonance that could not be easily explained. However, other works may include such instances and the issue of dissonance treatment may need to be revisited. 47 Scale-step theory is defined by Robert Wason as “a system in which scale degree chords are never subsumed by higher level phenomena.” See Robert W. Wason, Viennese Harmonic Theory from Albrechtsberger to Schenker and Schoenberg (Ann Arbor: UMI Research Press, 1985), p. 50. All these principles, now addressed in some undergraduate theory textbooks, such as in The Complete Musician by Steven Laitz, stem from the Stufentheorie of Gottfried Weber, who defined the role of a harmony by its placement in the key and standardized roman numerals as musical symbols in 1817. See Steven G. Laitz, The Complete Musician: An Integrated Approach to Tonal Theory, Analysis, and Listening, 3rd ed., (Oxford: Oxford University Press, 2012); Gottfried Weber, Versuch einer geordneten Theorie der Tonseztkunst, trans. James F. Warner (Boston: Oliver Ditson, 1846). 48 Cohn gives an example of such an instance, where in Wagner’s Tristan und Isolde, “the ‘seventh’ of the chord does not resolve, nor does it feel any pressure to do so. It is the putative root that is transient: as an under-seventh, it passes stepwise upward to the third of the resolution chord, just as a dominant seventh typically behaves, but in the reverse direction.” See Cohn, 143.

CHAPTER FIVE MAUS AND THE METER CYCLE: THREE NARRATIVE ANALYSES BRENT YORGASON

In his influential 1988 article “Music as Drama,” Fred Maus criticized music theorists for limiting the scope of their analyses to the description of musical structure, while neglecting other matters such as the emotive qualities of music or “the nature of the musical experience.”1 He also lamented the reliance on highly-technical language in analysis, citing Peter Kivy’s view that such description “leaves a large and worthy musical community out in the cold. Music, after all, is not just for musicians and musical scholars, any more than painting is just for art historians, or poetry for poets.”2 In an attempt to remedy this situation, Maus proceeds to demonstrate how correct analytical observations—which he calls “the solid achievements of theory and analysis”3—can be presented within a more inviting narrative that describes musical events in terms of agents and dramatic actions. In his analyses, he freely mixes theoretical language with anthropomorphicallyevocative dramatic language; ascribing thoughts, beliefs, desires, and other psychological states to musical agents in describing how they interact with and respond to one another. Dramatic actions can be associated with abstract musical events, or they might be attributed to fictional characters, including “fictionalized versions of the composer or performers.” Here’s how Maus describes it: “In listening to a piece, it is though one follows a series of actions that are performed now, before one’s ears . . . and in following the musical actions, it is as though the future of the agent is open—as though what he will do next is not already determined.” This necessarily involves “some kind of imaginative activity,” since the future actions of the performers are indeed “already prescribed.”4 These actions take place within a unifying plot that provides structure for the experience. But for the average audience member, the presentation and development of the plot matters more than its overall structure. The audience

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may already know the main plot points and perhaps even know how the story ends, so they are there principally for the experience. Similarly, Maus’s analyses are not concerned with a retrospective investigation of structure, rather they describe what is happening now in the development of the musical plot. In other words, his perspective reflects the active experiencing of music rather than a final understanding of it. His perspective suggests a particular position of observation within what I call the “meter cycle”—the series of phases through which meter passes as it moves from abstract notation to performance to experience (as depicted in Figure 5-1).5 If we were to envision the entire life-cycle of a work’s meter, we might find the series of activities depicted here: a work’s metric organization is initially determined by the composer, then notated in a score, performed by musicians, and heard and intuited by listeners (including composers, who apply this experience to future compositions).6 Like water in the water cycle, our descriptions of a work’s meter will change as it passes through the various phases of the meter cycle. The meter that is found “frozen” in the notated score must be thawed out and thoroughly melted by performers. A fluid performance of this meter then travels like metric vapor through the air to the listener. As listeners begin to intuit this aural meter, it condenses in their minds, becoming a very fluid metric wave. Then, as the experience becomes a memory, it solidifies in their minds once again. Continuing with this water cycle analogy, if we were to ask a group of scientists to describe the physical characteristics of water at various stages, we would get a variety of answers, ranging from “cold and hard” to “fluid and wet” to “warm and vaporous.” Similarly, our position of observation in the meter cycle will determine the characteristics that we find in meter.7

Figure 5-1, Phases of the “Meter Cycle”

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Figure 5-2, Detailed version of the meter cycle

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In Figure 5-2, I have fleshed out this cycle in more detail.8 Note that the circular flow of Figure 5-2 is the same as in Figure 5-1. Within this cycle, there are four different positions of observation that one could take. These are numbered in Figure 5-2 as:  a notated meter,  an expressed meter,  an internal meter, and  an abstract meter. Two of these positions (1 and 4) are retrospective positions of observation. Theories describing meter in these phases concern themselves primarily with abstract metric structure, from a viewpoint that is somewhat removed from the temporal experience. The other two positions of observation typically lead to processive theories of meter, with the observer positioning themselves within the flow of temporal events in time. This is the role that I imagine Maus would take, going back and forth between the patterns that are presented in the sounding surface of rhythms (the expressed meter at point 2) and the mental processes that interpret these patterns metrically (the internal meter at point 3), focusing more on the subjective experience of meter than its final structure. From a retrospective standpoint, the expressive elements found in the presentation of a work’s meter have little bearing on one’s conception of metric structure. Subtle variations in expressive timing and asynchrony are smoothed out or forgotten, and exceptions are eliminated in favor of uniformity. But from a processive perspective, these expressive performance timings do matter, since they are a part of our experience of the work’s meter. If we want to describe the experience of meter in a narrative analysis, we are going to focus more on how we experience beats and how the rhythmic events in a piece affect that experience.9 In this chapter, I will present three narrative analyses that use meter and expressive timing as the principal basis of the plot, with a particular focus on expressive asynchrony. In each of these analyses I will attempt to convey the “nature of the musical experience” by describing temporal events in terms of an ever-unfolding plot with agents that act and react. Each of the passages that I will examine can be seen as a type of dialogue—a dramatic scenario involving two main characters that sometimes cooperate but often compete with one another. In the first piece, Brahms’s Capriccio op. 76, no. 1, the competing agents are the left and right hands of the pianist. The principal melodic idea (sol-le-do-ti) is first introduced in the right hand in a stable metric context (see Example 5-1a). Twelve measures later, the same melodic idea is expressively delayed (see the boxed pitches in mm. 26–27 of Example 5-1b). Here, Brahms notationally displaces the melody from the barline by a sixteenth note, but the aural effect is likely one of expressive hand-

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Example 5-1a, Brahms, Capriccio, op. 76, no. 1, mm. 14–15

breaking—a performance technique which was typical not only of the late Romantic period but of Brahms’s playing as well—in which the right hand trails slightly behind the left.10 As a listener, I naturally attempt to hear the melody here in the same metric context as the previous non-displaced version, and am aided in this by the sfp (sforzando piano) marking given to the first note of the melody (versus a piano marking for the bass note). In his 1962 performance,11 Walter Klien slows down a bit at this phrase boundary, loosening my downbeat predictions and making it quite easy to latch onto the delayed upper note as the arrival of a “fuzzy downbeat.” But as this passage continues, an interesting dialogue begins to take place between the two hands. They now exchange thematic material every few measures, with each of the right-hand statements being expressively delayed, while the left-hand statements occur right on the notated beat (see the boxed pitches in Example 5-1b). Their conflicting metric orientations communicate two quite different temporal characters: the melody wants to linger on, dreamily floating behind the beat, but the lower voice continually interrupts it with an insistent urgency, as if to say “don’t slow down!” Whenever the bass enters, we must accelerate our placement of the downbeat, and when the soprano doesn’t immediately follow, we must wait for it to arrive. The recurring attentional shifts that I experience in placing the beat while listening to this passage are represented by the zigzagging arrows shown in Figure 5-3.12 In m. 36, the two hands present their conflicting beat possibilities simultaneously, rather than taking turns. This creates an agitated passage in which it is no longer clear whether the left hand is early or the right hand is late. In the score, the conflict between hands is abruptly brought to an end with the transformational synchronous downbeat of m. 38.13 But in Walter Klien’s performance, there is some additional cushioning provided here. Instead of playing the downbeat in m. 38 with hands solidly together, he again breaks his hands slightly, using decreasing amounts of

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asynchrony as the passage winds down, arriving at a perfectly synchronous downbeat only in m. 41. In other words, the performer provides a more nuanced expressive solution to the notational conflict set in motion by the composer.

Example 5-1b, Brahms, Capriccio, op. 76, no. 1, mm. 26–42

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Figure 5-3, Brahms, Capriccio, op. 76, no. 1, graphic representation of synchrony/ asynchrony

My second analysis describes a piano piece by Tchaikovsky that is aptly entitled Dialogue (see Example 5-2). The piece is clearly inspired by an operatic duet, with an animated discussion taking place between two lovers (a soprano and a bass). Tchaikovsky makes frequent use of notational displacement in this piece, possibly to imitate the effects of classical rubato that he had experienced in the opera house.14 Much of “Dialogue” is metrically quite hard to pin down for the listener, with irregular rhythmic patterns, unexpected harmonic shifts, and voices entering on weak beats as though they were strong. The overall effect is that of rhythmically-free vocal improvisation (marked quasi parlando, or “like speech”) over a fairly neutral chordal accompaniment. The narrative description I will provide here is just one of many possible interpretations of the dialogue taking place in this imaginary operatic scene. In the earlier part of the scene (see Example 5-2a), the two lovers casually engage in small talk, exchanging brief comments about nothing in particular. The bass’s replies to the soprano’s increasingly elaborate ideas are fairly limited expressively (each being a variant of the same descending half-step idea from G to F#—something like a musical “mmhmm”), but show that he is at least listening and moderately interested.

Example 5-2a, Tchaikovsky, Dialogue, op. 72, no. 8, mm. 1–10

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In Example 5-2b, the man finally gathers enough courage to pose “the lover’s question” (raising his dynamic level from piano to mezzo-forte for the first time in m. 19), to which the woman responds positively with a loving echo (m. 21). The man then resolutely declares his intentions with a descending mi-re-do line over the first clearly-stated cadence in the piece (m. 22). The soprano is clearly excited by this proposal, straying slightly from the steady accompaniment in a fit of passion (an agitato passage of metric displacement) in m. 23. The man, who remains grounded in the notated meter, tries to calm her (m. 26), but her passionate response continues until she submissively realigns herself with the accompaniment in m. 30. At this point the bass proclaims his undying love to the soprano in his most lyrical and sustained melody so far (see the passage marked dolce espressivo in Example 5-2c). To show his sincerity, he hesitates slightly (with sixteenth-note delays on each downbeat) but lingers sweetly on each accented syllable, conveying a sense of earnest longing and yearning. The woman listens carefully, with an occasional interjection of approval.

Example 5-2b, Tchaikovsky, Dialogue, op. 72, no. 8, mm. 19–28

Example 5-2c, Tchaikovsky, Dialogue, op. 72, no. 8, mm. 31–35

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Example 5-2d, Tchaikovsky, Dialogue, op. 72, no. 8, mm. 43–51

Then, in m. 39, the lovers—who up until this point have just been reacting to one another—begin finally to sing together; at first tentatively, then “un poco animando” (see Example 5-2d). Yet initially their love duet is not quite together; the woman’s voice trails the man’s voice slightly throughout (with music that is notationally displaced from the beat by a sixteenth note). The temporal friction between them becomes more heated as their voices rise in pitch and the accompaniment crescendos towards a climax labeled fortissimo appassionato. Here, at the climactic highpoint (in m. 49) they at last achieve perfect synchrony, gushing out their impassioned strains of love in perfect octaves.15 The work concludes with a series of gentle sighs and moans. My third analysis comes from Schumann’s Davidsbündlertanze. Like most of the music written around this time (1837), these pieces were influenced by Robert’s budding relationship with Clara Wieck, whom he married three years later. Schumann wrote to his former professor, Heinrich Dorn, that “the Davidsbündler dances were . . . almost entirely inspired by her.”16 These pieces express all of his passions, anxieties, longings, and dreams of eventually uniting himself with Clara. The piece I will be examining here is entitled “Wie aus der Ferne,” (or “as from afar”), which Schumann signs with both of his pseudonyms: Florestan and Eusebius. I will describe the metric narrative of this tender piece in terms of an intimate conversation between Robert and Clara (see Example 5-3). Robert’s voice is indicated by the dotted lines in the example; Clara’s voice is indicated by the solid lines.17

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In the opening measures, the lovers exchange words in overlapping, imitative gestures. The somewhat unusual grace-note chord in m. 3 (and later in m. 11) draws our attention to Robert’s overlapping response with an asynchronous gesture, as if he were interrupting Clara before she can quite finish. Their discussion is accompanied by quiet offbeat pulsation, acting as a subtle preparation for more conspicuous metric dissonance later on. In m. 17, this offbeat pulsation thickens in texture to produce full chords; and the periodic bass-line articulations (circled in mm. 19, 22, and 23) realign themselves with the displaced accompaniment. These questioning strains (first stated by Clara, then echoed by Robert) convey anxiety and indicate growing conflict between the two lovers, a sentiment that is heightened by strong, constant syncopation and a shift to the parallel minor. In m. 26 there is an unexpected modulation to the remote, yet refreshingly peaceful key of F major. The offbeat accompaniment here shrinks back to a single voice, offering us a moment of respite from metric dissonance as Robert pauses for a moment to listen to Clara’s ideas. Then, in a flash of inspiration, Robert enters with an idea of his own; but it is slightly offbeat (entering an eighth note behind the established meter in m. 28) and the shifting registers in his reply give us a little glimpse into his split personality (the “inner voices” in mm. 31–33). Nevertheless, Clara patiently listens. Clara’s silence and an unexpected shift of the accompanimental offbeats to the notated on-beats in m. 28 cause us to give our full metric attention to Robert’s offbeat argument, enabling a moment of what I call metric drift.18 That is, we are led to believe that what we are hearing is a metrically stable melody with an offbeat accompaniment, when in reality the beat that we perceive is notationally displaced from the barline. So far, then, in Harald Krebs’s terms, a weak metric dissonance (the offbeat accompaniment at the beginning) has prepared the emergence of a more strongly felt dissonance through the process of intensification (the thickened offbeat sonorities).19 And now, our metric focus may be given entirely to this dissonant metrical layer. Yet it is the resolution of this dissonance that is the most remarkable aspect of this passage (and one not easily explained by Krebs’s methods) in that it involves the addition of time that should not properly exist in a notated measure. In m. 34, Schumann defies the stringent boundaries of the barline by sneaking in an extra eighth note. Although he notates this as a grace-note chord (in order to maintain theoretical “correctness”) it is an extra eighth note and an


Maus and the Meter Cycle: Three Narrative Analyses

Example 5-3, Schumann, Davidsbündler, op. 6, no. 17, mm. 1–35

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integral part of the musical argument. In Wilhelm Kempff’s performance, this grace note chord is clearly played as an eighth-note upbeat, thereby creating a #4measure with seven eighth notes in it. The fascinating result— as we transition smoothly into a unified downbeat in m. 35—is that while the lower voice sounds as though it had been (metrically) correct all along, the upper voice turns out to be right as well, since it is the notated meter to which both voices attune themselves from this point on. The solid entrance of Clara’s melody on the high F# seems to suggest transcendence over metric disagreement. Indeed, for two full measures, there are no offbeat accents (the only moment in the piece where this is the case), indicating that a temporal plateau has been reached—a moment of agreement between two shifting temporal planes.20 If indeed Schumann was envisaging his union with Clara in this piece, I propose that this blissful moment arrives at the climax in m. 35. His elation at the thought of this union is clear in the peaceful running notes of mm. 35–36. Additionally, note how the registral shifts of Schumann’s conflicting “voices” are brought into check with the parallel octaves in m. 34. Perhaps it is even the quieting of these inner demons that allows Robert and Clara to at last come together. One common thread running through these analyses is that what we see in the score as displacement can be experienced in a less “digital,” more “analog” way. Rather than having to reinterpret our sense of meter, often what we face in these situations is a kind of subtle “tugging” at the meter—a conflict, to be sure, but a subtle one of beat placement rather than metric displacement. As Wallace Berry puts it, sometimes the barline “wobbles … events momentarily ‘tug’ at the bar line one way or another and it is promptly reaffirmed.”21 In other words, the effects of expressive timing and expressive asynchrony in performance can “tug at” our metric attention, causing certain beats to “wobble” in our internal meter. Although this expressive dissonance does not fundamentally alter any underlying metric structure, it does alter our listening experience, often in beautiful and remarkable ways. As listeners, we can all engage with the expressive effects of timing and asynchrony. As analysts, we can either choose to look past these effects, or we can position ourselves, as I imagine Maus does, within that portion of the metric cycle that deals with temporal experience more than memory or abstract notation. And in doing so, we can create a dramatic narrative that speaks not only to music scholars, but to all sensitive music lovers.

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Fred Everett Maus, “Music as Drama,” Music Theory Spectrum 10 (1988): 56. Peter Kivy, The Corded Shell: Reflections on Musical Expression (Princeton: Princeton University Press, 1980), 8–9. 3 Maus, 60. 4 Ibid., 67. 5 The concept of a “meter cycle” is introduced and explained in Chapter 1 of my Ph.D. dissertation, “Expressive Asynchrony and Meter: A Study of Dispersal, Downbeat Space, and Metric Drift” (Indiana University, 2009). 6 I have positioned analysis as an offshoot of the metric experience, where meter is intuited by a listener. It should be noted, however, that analysis can occur at other phases along the cycle, such as in the examination of a notated score (“notated”) or in the examination of the timing data from a performance (“performed”). In either of these cases, analysis is the only activity that is not necessary to the continuation of the cycle. 7 The position of observation that I most frequently adopt as an analyst is “wet” or “vaporous” rather than “cold and hard.” That is, I am more interested in the fluid performance of meter and in the listener’s metric experience than in the metric structure itself. 8 I will not discuss all aspects of this diagram here. The diagram is described in more extensive detail in Yorgason, “Expressive Asynchrony and Meter,” 13–32. 9 It is worth noting here that most narrative analyses tend to focus on pitch—with notes, harmonies, and keys acting as the main agents. Narrative analyses describing expressive timing and the experience of meter are less frequently encountered. 10 See Will Crutchfield, “Brahms, by Those Who Knew Him,” Opus 2, no. 5 (1986): 18. This manner of playing can be heard in recordings by many lateRomantic pianists, even in passages that are notated to be played with hands together. 11 Walter Klien. Piano Music, Vol. II. Vox SVBX 5431. 1962. 12 This type of attentional diagram can be used to illustrate a listener’s “path of metric focus” as they experience a work’s meter in time, as described in chapter 7 of “Expressive Asynchrony and Meter” (2009). Note that different listeners may experience different paths of metric focus. 13 This initiates a slower passage featuring an augmented inversion of the principal melodic idea. 14 With classical rubato, the melodic right hand is rhythmically more flexible than the left, holding back or accelerating slightly while the accompaniment continues in strict time. Classical rubato likely originated in the opera house, as singers manipulated the timing of their phrases in relation to the orchestral accompaniment. See David Epstein, Shaping Time: Music, the Brain, and Performance (New York: Schirmer Books, 1995) and Richard Hudson, Stolen Time: The History of Tempo Rubato (Oxford: Clarendon, 1994) for more information about this type of rubato. 2

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15 Note that the male character has somehow been transformed into a “heroic tenor” here, reaching a high B4 in measure 49 (notated in the upper staff). 16 The Letters of Robert Schumann, ed. Karl Storck (London: John Murray Press, 1907), 131. 17 There are also some thin vertical dotted lines in a few places, showing how Wilhelm Kempff aligns events between hands in his performance on Klavierwerke / Robert Schumann. Deutsche Grammophon 435 045-2. 1991 [recorded 1967–73]. 18 Metric drift can be described as a situation in which the listener’s placement of beats drifts slightly as their attention is drawn to a metric stream that very subtly pulls them away from the notated beat. See Chapter 6 of Yorgason, “Expressive Asynchrony and Meter” for a more detailed discussion of metric drift. 19 See Harald Krebs, Fantasy Pieces: Metrical Dissonance in the Music of Robert Schumann (New York: Oxford University Press, 1999), 97. This type of intensification is also referred to by Krebs as “surfacing”—a process by which a subliminal dissonance becomes more prominent and moves to the fore. 20 The transformative nature of this moment is also reflected in the harmonic language: note how the G7 in measure 34 is enharmonically reinterpreted as a Ger+6 leading to a glorious “arrival six-four chord” in the home key of B major. 21 Wallace Berry, “Metric and Rhythmic Articulation in Music,” Music Theory Spectrum 7 (1985): 13.

PART II: MUSIC OF THE TWENTIETH CENTURY

CHAPTER SIX VOICE LEADING AND MUSICAL SPACES IN BRITTEN’S OPUS 70 DALE T. TOVAR

Britten’s Opus 70 (Nocturnal) can be formally understood as a reverse theme and variations for which the theme is Come, Heavy Sleep. The lutesong Come, Heavy Sleep was a poem that John Dowland set to music, and it appears as song no. 20 in his acclaimed “The Firste Booke of Songes (sic)." Though Come, Heavy Sleep is “tonal"1 and there are many tonal–like aspects and pitch–centric passages in Nocturnal, Nocturnal is not tonal. This demonstrates Britten's philosophy that the present (reflected in the musics of the modern classical guitar and modern compositional techniques involving octatonic collections and interval cycles) must not merely appreciate the past (reflected in the musics of the lute, medieval poetry, and methods of traditional harmony), but understand its singular role in its creation. Indeed, it is well known that Britten worked for much of his life on a revival of British music and so the choice of a song by Dowland and a musical form resembling a classical form was appropriate, considering his goal. There have been repeated attempts to explore the relationship between the highly chromatic foreground and middleground material, and the theme. For example, Becker (2012) theorizes that there are middleground pitches one semitone apart that are continually paired against each other. He calls these ic1 pairings and finds ic1 pairings related to each other by T4, creating a background hexatonic collection. Also, he finds that some of the foreground can be segmented into sets whose set classes contain inversional symmetry. He demonstrates his findings primarily using Musingly, the first variation of Nocturnal. Becker provides a wellgrounded middleground analysis but his method for analyzing foreground material accounts for only a relatively small amount of the foreground, and he shows no relationships between the sets he identifies. Lee (2013) explains the chromatic foreground material primarily in terms of

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polytonality. However, to a lesser degree he also analyzes the material in terms of octatonic and whole-tone collections. While Lee shows an excellent “Map Analysis" detailing the melodic relationships between Musingly and Britten's transcription of Come, Heavy Sleep and this is the main focus of the analysis, he has no explanation of any middleground and makes tonal claims that are difficult to justify with the notes given and, in some passages, offers no definitive interpretation of the harmonic movements. Rupprecht (1996) argues that in Musingly there are two prominent middleground registral contrapuntal lines. This approach of identifying a middleground structure with registrally related pitches falls under what Straus (1987) and (1997), calls the “associational” model. Boss (1994) writes “This practice of calling pitches structural because of factors – sometimes rhythmic, dynamic, and timbral – that cause them to stand out from other pitches in their immediate contexts is common to every writer on structural levels in non-tonal music." Straus (1997) identifies three prominent voice-leading models in atonal music: prolongation, association, and transformation. Despite this logical identification, associational and transformational analyses describe different phenomena; transformational voice leading (as used by Lewin (1998), Klumpenhouwer (1991), O'Donnell (1997), and Straus (1997), (2003), (2005), and (2014)) describes the relation between two pitch–class sets (pcsets) in pitch–class (pc) space while association describes the contextual relationship between objects (pitches, motives, or chords) in a piece of music. For example, saying that some chord x occurs in the same register as chord y does not tell us if x and y are similar chords or not. Conversely, saying that T5: x → y, tells us how the chords are related in pitch–class space but not in the context of the piece. In this way, the theories are complementary. Most all of the middleground pitches in Nocturnal are associated by register in pitch space as well as by a consistent pitch interval. The notation used for a pitch interval cycle will be c ± x. Octatonicism is an aural technique used by many composers such as Debussy, Stravinsky, and Britten. The role of octatonicism in the music of Stravinsky has been one of much debate. Tymoczko (2002) writes, “I do not question that the octatonic scale is an important component of Stravinsky's vocabulary, nor that he made explicit, conscious use of the scale in many of his compositions. I will, however, argue that the octatonic scale is less central to Stravinsky's work than it has been made out to be." One of the issues that van den Toorn (2003) raises with Tymoczko’s argument is that “No mention is made of the three transpositions of the octatonic set (Collections I, II, and III), of the two possible scales or

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interval orderings, and of the explicit ways in which these scales interact with various diatonic scales in Russian and neoclassical works." Regardless of their specific role in the music of Stravinsky, octatonic collections and the interactions between collections play a large role in Nocturnal. In this discussion, OCT0,1, OCT1,2, and OCT2,3 will be the designations for the octatonic collections, as shown in Example 6-1. In Nocturnal, though entire iterations of the octatonic scale are present, the use of octatonic collections is primarily employed through subsets. The subsets most commonly used are the set classes (013), (0136) and (01346). These are almost always ascending and take the form of the pitch interval motives , , and .

Example 6-1, The three octatonic collections. The pcset {0134679t} is designated by OCT0,1. Likewise, {124578te} by OCT1,2 and {0235689e} by OCT2,3.

The interaction between octatonic collections can be categorized into three motions from an arbitrary collection: to the same collection, to the collection related by T1, and to the collection related by T2. I propose the octatonic GIS (S, G, int) where S is the collection of pcs {0, 1, 3, 4, 6, 7, 9, t}, G is the integers under addition mod 8, and for s, t
S the function int(s, t) = “number of hours clockwise from s to t"2 on an 8-hour clock. As we are comparing different octatonic collections, the group G as well as the function int will be the same for all GISes discussed. We can define each of the three octatonic collections as its own GIS: S0 = {0, 1, 3, 4, 6, 7, 9, t}, S1 = {1, 2, 4, 5, 7, 8, t, e}, and S2 = {2, 3, 5, 6, 8, 9, e, 0} where Sn denotes the space because they share the group G. Because |S0| = |S1| = |S2|, we define the GIS isomorphism3 OT where OT: S0 → S1 where for x
S0, OT1(x) = x + 1 mod 12
S1. Here we have a way of expressing the interactions between octatonic collections. This way of modeling a change of collection has several advantages. For instance, the PC set {013467} cannot be mapped onto {23568} using traditional transposition or inversion. Instead, if we see these objects in distinct spaces, we can retrieve a musically meaningful relationship. A spatial network of octatonic collections using OT1 is shown in Example 6-2.

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Example 6-2, A spatial octatonic network

The whole-tone scale also plays a role in the foreground of Nocturnal. In this discussion the collection (02468T) is labeled as WT0 while the complementary collection, (13579E), is WT1. A sequence containing multiple iterations of material consistently transposed either by T2 or Tt is suggestive of a whole-tone collection. In Nocturnal, the material that is sequenced is often from an octatonic collection. Sequences with a consistent Tn form an interval cycle. Interval cycles play a large role in both the foreground and middleground of Musingly, as will be shown. Such cycles facilitate the prominent OT1 mapping between octatonic collections. *** For the purposes of this study, we understand a voice V as a sequence of notes. A note n will be defined as the triple (pitch–class, octave number, rhythmic value). An “event” e is the set of all notes beginning at the same time interval. Lastly, we can define a “passage" as the sequence of all events in a given time–span in a piece of music.4 More properly, a Voice is a subpassage of subevents. This definition of a voice translates easily to a “2-dimensional array" in higher-level programming languages. Example 6-3 shows the necessary Java code to determine the longest and highest notes in some arbitrary passage. These notes can be collected into a “salientVoice." Without diving much further into this, other contextual criteria go into whether a voice element is salient or not. As was mentioned, interval cycles and whole–tone collections govern the pc connections between distant voice elements. So, if the interval between notes in the salientVoice (determined by “for structures" of contextual criteria) is a constant, both notes belong to the middleground voice. Likewise, if a note in salientVoice belongs to WT1, the note belongs to the middleground voice or Voicemid.

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Note highest = new Note (0, 0, 0) for (int i = 0; i < passage.length; i++) { for (int j = 0; j < passage[i].length; j++) { if (passage[i][j].oct > highest.oct) highest = passage[i][j]; else if (passage[i][j].oct == highest.oct && passage[i][j].pc > highest.pc) highest = passage[i][j]; } }

Note longest = new Note(0, 0, 32) for (int i = 0; i < passage.length; i++) { for (int j = 0; j < passage [i].length; j++) { if (passage[i][j].rhythm < longest.rhythm) longest = passage[i][j]; } }

Example 6-3, Necessary Java code to determine the highest and longest values

(a) for whole–tone collections: if (salientVoice[i][j].pc % 2 == 1)

(b) for interval cycles: if (salientVoice[i-1][j].pc - salientVoice[i][j] == n)

Example 6-4, The segregation rules of the middleground voice.

Nocturnal is made of eight variations and the theme, based on Come, Heavy Sleep, comes at the end. The first variation, Musingly, is explored phrase by phrase in detail here. The middleground of Musingly will be examined as well. For the most part, breath marks denote ends of phrases. The last measure of the fourth system and all of the fifth system is a single phrase. The opening phrase of Musingly found in Example 6-5 features two octatonic collections branching out from A. The pcs G#, A, B, and C make up an ascending form of sc (0134) that belongs to OCT2,3. The notes A, G, and F# make up a descending form of sc (013) that belongs to OCT0,1. OT1 maps OCT2,3 onto OCT0,1 and is shown in Example 6-5. The notes returned by the salience method are {A, F, F#}. As there is yet to be a reoccurring interval pattern, A, the most prominent of the notes, is the only note segregated into Voicemid.

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Example 6-5, the first phrase of Musingly. © Copyright 1965 by Faber Music Ltd, London. All score excerpts are reproduced by kind permission of the publishers.

The second phrase shown in Example 6-6, like the first, contains two octatonic collections. In this case, there are two forms of sc (013). Opposite of the first phrase, the first form of sc (013) descends with the notes B, A#, and G#, and belongs to OCT1,2, while the other form of sc (013) ascends and contains the notes A#, B# and C#, and belongs to OCT0,1. Unlike most of the piece which uses OT1, OT2 is used to map OCT1,2 onto OCT0,1.

Example 6-6, the second phrase of Musingly

The first notes of the first two phrases are A and B respectively, with both phrases containing two forms of sc (013), one ascending, the other descending from distinct octatonic collections. This gives the relationship a T2-like quality. The B then leads into the C# at the end of the second phrase making the middleground motion of the first two phrases a c+2. Here, the B is formally parallel to the A and is in salienceVoice. The C in m. 2 fails to conclude the phrase on an upward motion. The C# in m. 4 succeeds in this. Now, Voicemid = {A, B, C#}. Not only does each element belong to WT1, but also a reoccurring interval pattern has been established. The third phrase, seen in Example 6-7, opens with a form of sc (01346) starting on the note A and belonging to OCT2,3. The first three notes of both measures six and seven are forms of set class (024) leading into sequenced forms of sc (013) in which OT1 maps the spaces of adjacent forms of (013) to the next. The forms of sc (013) are sequenced at c–2. In

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each case, the motive takes the form of . After the sequence ends, the note B is repeated before following with A, G, and F, which makes a c–2 of WT1. The highest note of each form of sc (013) also is a member of the c–2. Each of these notes are elements of Voicemid. m. 5

6

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Example 6-7, the third phrase of Musingly

In the first two phrases, Voicemid is comprised of a c+2 containing the notes A, B, and C#. In the third phrase, Voicemid is comprised of a c–2 starting on Eb, which is the following member of the c+2 in the previous phrases. The next phrase (Example 6-8) begins by repeating the notes A, G, and F before continuing the c–2 with the notes Eb and Db, though the Db is not salient. The notes that follow are C, D, Eb, F, and F#, which is a form of sc (01346) belonging to OCT2,3. The note G occurs before the F# as a member of the G, F#, and E form of sc (013) that belongs to OCT0,1. This is yet another instance of OT1.

Example 6-8, the fourth phrase of Musingly

The fifth phrase, shown in Example 6-9, begins with nine ascending sixteenth notes leading into the note B. The first four notes, E, G, A, and Bb, are a form of sc (0136) which belong to OCT0,1. The next six notes, D, E, F, Ab, Bb, and B, are a form of sc (013689) and belong to OCT1,2, maintaining the use of the OT1 shown in Example 6-9. After the B, a sequence of forms of sc (013) transposed down two semitones starts, including a form made up of the Ab, Bb, and B from the form of sc

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(013689). This is similar to the sequence in the third phrase in Example 67 in which OT1 is used repeatedly. After the G, a form of sc (013) not belonging to the sequence takes place with the notes C#, D#, and E, but also belonging to OCT0,1. The E is then followed by D and C, making a form of sc (024) that belongs to WT0.

Example 6-9, the fifth phrase of Musingly

Up to this point in the piece, all of the middleground notes can be associated by their relationship in pitch space. The Db from the fourth phrase now changes registers as it moves to the B in the fifth phrase. The notes B, A, and G are the top notes of each form of sc (013) and are clearly emphasized metrically and by duration, and continue Voicemid. The sixth phrase (Example 6-10) begins on an E major triad with the melody note F on the top. This F is part of Voicemid. Then a C enters and is alternated with the F. The E major triad with the C on the top belongs to sc (0148). A form of sc (01346) begins with the notes C, D, Eb, F, and Gb, which belong to OCT2,3. The phrase ends on an A major triad with an F on top, which also belongs to sc (0148). The Eb continues Voicemid which comes to an end on C# as the highest note in the A major arpeggio. All of Voicemid up to this point is shown in Example 6-11.

Example 6-10, the fifth phrase of Musingly

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Example 6-11, Voicemid through the first six phrases of Musingly

The next phrase is comprised solely of forms of sc (01346) at different transpositions. Set class (01346), as a subset, is a transpositional generator of the octatonic collection at T6. Britten begins the phrase in OCT1,2 with the notes F, G, Ab, Bb, and Cb. This is then transposed down a perfect fourth to start on C. After C, D, Eb, F, and Gb, the (01346) is transposed to start on F#, completing an octatonic scale of OCT2,3, which is an OT1. The same transposing down a perfect fourth and then up by a tritone occurs, making an OT1 cycle, all of which is shown in Example 6-12.

Example 6-12, the seventh phrase of Musingly

The eighth phrase of the variation begins with a form of sc (013469) from OCT1,2 containing the notes E, C#, B, A#, G#, and F-double sharp. The F-double sharp also belongs to a form of sc (02358) from OCT0,1 with the notes F-double sharp, A#, C#, D#, and C, ending the phrase (Example 6-13). The motion from OCT1,2 to OCT0,1 is OT2. The middleground motion of the phrase is from C# to C.

Example 6-13, the eighth phrase of Musingly, bringing back the unusual motion from OCT1,2 to OCT0,1 that was used in the second phrase.

Example 6-14 shows the middleground pitches from the seventh and eighth phrases. The first five pitches are from the seventh phrase and are

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the first notes of each of the iterations of sc (01346). The C# remains as the first note in the eighth phrase and moves back down to the C.

Example 6-14, the middleground pitches from the seventh and eighth phrases

The last phrase of Musingly shown in Example 6-15 is nearly identical to the sixth phrase shown in Example 6-10. The C from the previous phrase then ascends up a perfect fourth to F with an E major triad harmonized under it. The C and F are alternated until the variation ends on C.

Example 6-15, the last phrase of Musingly

*** Though not identical, the phrases of different variations are related to each other. And so, the phrase structure of each of the variations, except the Passacaglia, is nearly the same. The material from the sixth phrase is repeated in the last phrase of Musingly. Likewise, in Very Agitated, the material from the sixth phrase is repeated in the last phrase. Most all of the sixth phrases share an important feature. In Musingly, the open low E string is not played until this phrase where there is a pedal E for all but the last chord. In Very Agitated, outside of chords at the ends of phrases, the low E is not played until this phrase. The corresponding phrase in the third variation also possesses this feature. Also, the motive of the melodic perfect fourth is emphasized in each of these phrases. Though the Passacaglia does not follow the same phrase structure as the others, there is a pedal E present throughout. Come, Heavy Sleep is played at the end of Nocturnal in the key of E major.

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Example 6-16, an event network of all of Musingly in terms of GIS isomorphisms between octatonic collections

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The variations of Nocturnal After John Dowland are seemingly very different from one another, but they share a model phrase structure. The use of interval cycles is ever present among the variations as well as use of octatonic collections and the prominent OT1. The way in which whole octatonic collections are implied by their subsets and the way that Britten moves between octatonic regions helps to define this piece. The interest here with cross-type transformations is to see how different spaces move in a consistent manner, as is shown in Example 6-16. By using an analytically formed method of justifying that certain notes are elements of a middleground voice, we are able not only to see how a whole-tone interval cycle is realized in Musingly, but also how these motions relate to the chains of OT1.

Appendix A The Java code produced in Examples 3 and 4 relies on classes for notes, events, and passages. The basic variables and constructors are included in the following. public class Note { private int pc; private int oct; private int rhythm; public Note (int pc, int oct, int rh) { this.pc = pc; this.oct = oct; rhythm = rh; } } public class Event { private Note [ ] event; public Event (Note [ ] event) { this.event = event; } } public class Passage { private Event [ ] pass;

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public Passage (Event [ ] P) { pass = P; } }

Notes 1

By “tonal,” I mean possessing pitch–centricity and the prominent use of diatonic collections. To a modern listener, Dowland’s piece certainly sounds tonal. However, as tonality had not been invented technically in Dowland’s day, it is safe to assume that Dowland did not see it in this perspective. 2 Lewin (1987). 3 Hook (2007). 4 Tovar (2015) provides much more extensive definitions for voice, note, and passage. However, for this study, the definitions given here are suitable.

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Works Cited Becker, Thomas. 2012. “Analytical Perspectives On Three Groundbreaking Composers for Guitar: Villa-Lobos, Martin, and Britten.” PhD diss., University of Kansas. Boss, Jack. 1994. “Schoenberg on Ornamentation and Structural Levels.” Journal of Music Theory 38 (2): 187-216. Hook, Julian. 2007. “Cross-Type Transformations and the Path Consistency Condition.” Music Theory Spectrum 29 (1): 1-39. Klumpenhouwer, Henry. 1991. “A Generalized Model of Voice-Leading for Atonal Music.” PhD diss., Harvard University. Lee, Andrew. 2013. “Charlatans, Saltimbanques and Lumpen Songs: The Emergence of the Guitar from the Ghetto of Western Art Music. PhD diss., Princeton University. Lewin, David. 1987. Generalized Musical Intervals and Transformations. New Haven: Yale University Press. —. 1998. “Some Ideas About Voice-Leading Between PCSets.” Journal of Music Theory 42 (1): 15-72. O'Donnell, Shaugn. 1997. “Transformational Voice Leading in Atonal Music.” PhD. diss., Graduate Center of the City University of New York. Rupprecht, Philip. 1996. “Tonal Stratification and Uncertainty in Britten's Music.” Journal of Music Theory 40 (2): 311-46. Straus, Joseph. 1987. “The Problem of Prolongation in Post-Tonal Music.” Journal of Music Theory 31: 1–22. —. 1997. “Voice Leading in Atonal Music.” In Music Theory in Concept and Practice, ed. James Baker, David Beach, and Jonathan Bernard, 237–74. Rochester: University of Rochester Press, 1997. —. 2003. “Uniformity, Balance, and Smoothness in Atonal Voice Leading.” Music Theory Spectrum, 25 (2): 305–352. —. 2005. “Voice Leading in Set-Class Space.” Journal of Music Theory Vol. 49 (1): 45-108. —. 2014. “Total Voice Leading.” Music Theory Online 20.2. Tovar, Dale T. 2015. “Hearing Voices: Reconciling Theories of Structure and Voice Leading and Their Application to Atonal and Other Music.” Undergraduate thesis, Eastern Oregon University. Tymoczko, Dmitri. 2002. “Stravinsky and the Octatonic: A Reconsideration.” Music Theory Spectrum 24 (1): 68-102. van den Toorn, Pieter C. 2003. “Stravinsky and the Octatonic: The Sounds of Stravinsky.” Music Theory Spectrum 25 (1): 167-85.

CHAPTER SEVEN ENDING LIGETI’S PIANO ETUDES SARA BAKKER

Repetition provides composers with a powerful source of unity and an approach to creativity through self-imposed limitations, offering an appealing alternative to tonality, serialism, or symmetry. Repetition can also, however, create significant aesthetic problems for the composer, such as how to keep repetition interesting in the short term and how to convey a sense of form in the longer term.1 This conflict is perhaps nowhere more apparent than in musical textures that are saturated with pattern repetition. Fundamentally, it is a conflict between repetition’s tendency toward continuity and on-goingness, and our aesthetic need for closure and resolution. What are the ways one can end a piece that is based on pattern repetition? Is a convincing ending even possible, or will the piece simply have to stop?2 György Ligeti provides two solutions in his piano etudes. Although much is known about Ligeti’s fascination with repetition in his middle-period works,3 analytical scholarship on his more recent music instead tends to focus on other features. For example, scholars have theorized about the composer’s stylistic influences,4 use of irony and parody,5 and use of nonfunctional harmony.6 While it is certainly true that Ligeti’s late style introduced many provocative features, this paper asserts that repetition remains an important creative generator in Ligeti’s late music and examines the existential conflict between local repetition and largescale closure in two Piano Etudes: Désordre (1985) and En suspens (1994). My approach to rhythmic repetition draws on the cycle, an analytical unit defined by the limited number of ways in which strictly repeating patterns can coincide. The cycle is introduced in Gretchen Horlacher’s (1992) analysis of repetition in the music of Igor Stravinsky.7 Her goal in this study is to show that repetition is not antithetical to concepts of musical development, that the variety of metrical contexts each pattern sounds in as well as subtle changes to the patterns themselves prevent an overly rep-

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etitious texture. In many cases, the cycle defines phrases, although it can also be “frustrated” in this sense by a change in the pattern and then continued in the following phrase. Example 7-1 outlines a simple cycle drawn from Horlacher’s study. It consists of two repeating patterns, rhythmic ostinati, equivalent to three half notes in the right hand and four in the left. Each time the right-hand ostinato starts over, it sounds against a different beat in the left: first with beat 1, then 4, 3, and finally 2. Example 7-1 stops just before the cycle would return to its original contrapuntal relationship, where the beats 1 would coincide. At this point, once the cycle has realized all of the ways in which the ostinati can come together, it is complete.8

Example 7-1, Complete cycle, after Horlacher (1992), Adapted from Stravinsky, Symphony of Psalms, III

Inherent to the definition of the cycle are a predetermined number of unique contrapuntal combinations, which I will call alignments after Sara Bakker (2013).9 Alignments name a specific point in the cycle, lasting from one start to the next in a single ostinato. In Example 7-1, alignments are labelled according to the beats that sound together and placed on the score, circled, in the middle of the staff. In order to prevent multiple names for a single alignment, I show them only where the upper part starts over, which will always result in alignments with a “1” on top. The number of alignments in a given cycle, that is, the number of alignments required to complete the cycle, can be calculated using the “least common multiple” of the ostinati using some shared note value. For the cycle in Example 7-1, the calculation would look like this: LCM (3, 4) = 12 This tells us that the cycle will be complete after 12 half notes, if both ostinati repeat strictly. In short cycles such as this, it is actually quite easy to see that the cycle is complete, that the next alignment would be a 1/1. Not all cycles are so concise, however, and in those cases, it is helpful to know the cycle’s length early in the analytical process as it can help predict where new alignments should be.

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As suggested above, Horlacher’s (1992) work on the cycle models the relationship between repeating ostinati and local formal boundaries, such as phrases. Complete cycles often define such events in the music of Stravinsky. In his repetition-based piano etudes, however, Ligeti experiments with different ways of using cycles. The repetition-based piano etudes are a diverse group of four pieces, each based entirely or in large part on the concurrent repetition of two or more ostinati. Two of the etudes, Fanfares and Fém (Nos. 4 and 8 respectively), complete a cycle at their outset and then incorporate small changes to the ostinati that greatly affect how they interact. Most of these pieces are a kind of compositional (and analytical) study in cycles that are not complete.10 The other two, Désordre and En suspens (Nos. 1 and 11), contain no complete cycles at all. In fact, complete cycles would be impossible within the durational limitations of the pieces; the cycles are simply too long. The focus of this paper is on the latter scenario and the existential conflict between form and content that arises in pieces that are conceived in repetition, but cannot rely on that repetition to come “full circle” and provide satisfying closure. I will discuss each piece in turn and then turn to some of the broader issues that this sort of conflict entails. Désordre consists of two related rhythmic ostinati, one in each hand. Although the ostinati are notated as a continuous stream of running eighths, accents and octave doublings group the eighth notes into beats that are contextually S, L, and D. Example 7-2 shows the ostinati on a single rhythmic line, using upper slurs for the right hand and lower for the left hand. The slurs highlight a sentence construction in the ostinati, one that is somewhat unconventional in its proportions. In the right hand, it is 4+4+6 (where the Abgesang is too short), while in the left hand, it is 4+4+10 (where the Abgesang is too long.) It is precisely the difference between these Abgesänge that makes the cycle possible; if the ostinati used the traditional 4+4+8 proportion, they would be the same length. The right-hand ostinato is 14mm long, while the left hand is 18mm. As it turns out, measure lengths are not consistent between the hands in this piece, and so the most precise way to discuss the ostinati is using eighth notes, the highest common multiple. I will still occasionally refer to measures, however, to give a better sense of scale.


Example 7-2, Rhythmic ostinati in  





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Closure through Cyclic Manipulation The cycle in Désordre is a superimposition of 109 eighths over 144 eighths. It has 144 unique alignments, each 14 mm. long, and unfolds over the course of 1962 measures in 8/8 time, over half an hour in performance.11 Interestingly, in addition to the regular changes in alignment here, Ligeti complicates the cycle by deleting one eighth note in every fourth measure of the right hand. Each hand continues to be barred according to its own ostinato, which results in split bar lines throughout much of the piece. It probably goes without saying that the cycle is impossibly long for so demanding a piano piece, both physically for the performer, but also mentally for the listener. Although there might not be any significant way to mediate listener boredom, there is, however, a long history of “impossible” music being successfully performed, so perhaps pianists would rise to the challenge. Nevertheless, Ligeti seems aware of the tensions here, and does not complete the cycle he introduces in Désordre. Instead, the piece as written is closer to 132 mm. (assuming 8/8 time), or just under 2.5 minutes in performance.12 If Ligeti does not complete the cycle, what does he do? How does the piece find satisfying conclusion? One way to approach such an issue, where the very essence of the piece is impossible within its own limitations—or at least in conflict with its practical realization—is to simply let the piece be, as a kind of contemplation of things beyond our reach. Another way is to try to resolve it in some way, and that is what Ligeti does in this case. Ligeti’s solution is to rework the cycle to make it fit, and he does so with the introduction of several “new” cycles, “new” because they are each clearly based on the original cycle. Example 7-3 provides a timelineoverview of the form of Désordre. Cycle alignments are listed below the timeline, and measure numbers are given independently for each hand (RH/LH), since they are not consistent in length. Example 7-3 shows five cycles in operation throughout the piece, none of which return or overlap. The first cycle takes up the most measures, but I will show that the others are equally important in this piece’s search for closure.




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Example 7-3, Formal overview of  


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Not apparent in Example 7-3, is the close connection between Cycles II – V and the opening cycle. They do not actually sound “new.” Each cycle uses the same melody, at a new pitch level for each repetition, and the same rhythmic profile of S, L, and D beats. Example 7-4 shows only the Stollen portion of each cycle, due to limitations of space, but represents the rhythmic profile of both hands unless otherwise indicated. Across the top of the example are the generic length descriptors of attacks in the Stollen: SL SL LS D. The small, greyed-out notes represent scalar connecting eighths that space out the main beats. On the far right, are some details about cycle length to facilitate comparison. Although the precise length of each attack type (S, L, or D) varies from cycle to cycle—Ds range from 2 to 9 eighth notes, for example—the relationship of S, L, and D beats within each is consistent. The use of the same melodic and rhythmic profile in each new cycle suggests that these “new” cycles are actually reimaginations of the original, attempts to come up with an ideal cycle. S

L

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* measures counted in LH ** measures assume 8/8 time

Example 7-4, Corrective cycles in   (  portion only)

Consistent with my interpretation of subsequent cycles as reworkings of the original is the fact that the overall length of each cycle is different. The far-right column of Example 7-4 lists the length of each cycle, first in measures and then also in minutes, to give you a better sense of scale. Because measures in this piece vary widely in length, the measure lengths listed here all assume consistent 8/8 time. Ligeti makes three successful attempts at a corrective cycle in Cycles II, III, and IV. Cycles II and III represent marked and progressive com-

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pressions of Cycle I. This is most notable in the length of their D beats. Ds are worth 7 eighths (8 in the left hand) in Cycle I, 3 in Cycle II, and just 2 in Cycle III. The connecting eighths are responsible in this piece for spacing out the main beats, and when some are deleted, the main beats seem to come frantically faster and faster. The deletions result in two cycles that are short enough to be completed in Désordre’s 132 mm. Cycle II is just over 47 mm., short enough to make more than 2 complete cycles in 132 mm., while Cycle III is shorter still. At just over 32 measures, it could make more than 4 complete cycles. So clearly, Ligeti is able to reinvent the cycle, based on the original ostinati, such that it could be completed within a performance of about 2.5 minutes. He does not complete these cycles either. Perhaps they are too different from Cycle I; perhaps their drastic shortening of the D beats is too extreme. Enter Cycle IV, the happy medium. This cycle is in many ways a return to the original cycle. It restores the durations of the S and L beats, and makes the D beats the same. The pace here feels much less frenetic than in Cycles II or III, and the fact that the hands (and bar lines) remain coordinated in this cycle makes it feel settled in a way that was not possible in Cycle I. Cycle IV is also corrective, in the sense that it represents a shortened version of Cycle I. Although longer than the previous corrective cycles, it is still only 126 measures, and would easily fit in Désordre’s 132 measures, perhaps allowing a short introduction or conclusion. In Cycle IV, then, it seems that Ligeti has found the perfect solution to his problem of repetition and closure. He has made a cycle that can be completed within a reasonable length of time, one that maintains most of the features of Cycle I. Interestingly, though, he does not complete this one either, and in fact, goes one step beyond, creating an “incorrective” cycle, Cycle V. Cycle V maintains the more relaxed pacing of Cycle IV, but reintroduces a mismatch between the right and left hand’s D beats. The right hand uses 8 eighths here against the left hand’s 9. The discrepancy between the Ds results in the hands (and bar lines) slowly becoming uncoordinated. The left hand falls progressively 1 eighth behind the right hand with each D beat. A similar left-hand lag was an important feature of Cycle I, although there it was achieved by making the right hand faster (7 eighths versus 8 in the left hand). So why, after arriving at Cycle IV, did Ligeti not finish out the piece with that cycle? Perhaps he felt aesthetically that he had arrived at his solution too late in the piece, three-quarters of the way through. Perhaps the idea of a left-hand lag was too fundamental to the piece to be omitted. Perhaps he realized another way to achieve closure, one not reliant on the

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cycle: I suggest that Ligeti achieves closure in Désordre through the very process of manipulating the cycle. Rather than completing every possible alignment that a given cycle entails to find closure, he instead exhausts the possibilities of manipulation. He starts out with a cycle that is much too long, so he shortens it. He goes too far, however, and then expands it. Again, he goes too far, but at this point the piece is ready to be done, and none of the cycles achieve closure through completeness. Through systematic manipulation, however, he achieves a different kind of closure. He come full circle through cycle manipulation, starting and ending with cycles that are impossibly long, achieving therein a kind of closure that was not suggested organically by the materials themselves.

Closure through Thematic Means En suspens is very different in its organization and overall effect from Désordre, but also shares important points of contact on an organizational level. Unlike Désordre, the tempo is quite slow in this piece, and there are no running eighths to subdivide the main beats until the final section of the piece. The hands are actually notated individually in 6/4 and 12/8, such that bar lines do line up, but beats do not. Still, there is not a strong sense of meter in this piece—many attacks are tied across the bar line, and there are additional, written-in accents that do not support a traditional experience of either notated meter. Similar to Désordre, however, En suspens uses a limited number of contextually defined beats. Here, they are Ls and Ss, although because of the different time signatures in the hands, Ls and Ss are only consistent within a given part and section. In section A, Ss in the right hand are worth 2 eighths and Ls, 4, while Ss in the left are worth 3 and Ls 6. Finally and perhaps most significantly, this piece is similarly organized by a cycle, but does not find closure through cyclic means. The cycle in En suspens consists of an ostinato in the right hand of 50 eighths, which sounds against one of 36 in the left hand. Example 7-5 shows the ostinati in generic notation, using Ls and Ss, in the outer sections of this piece, A and Aʹ. The piece opens by outlining a cycle that is 900 eighth notes (75 measures) long, lasting 5 minutes in performance.13 While this cycle may seem short compared to some of Désordre’s, this piece ends in less than half of that time: En suspens as published consists of 35 measures, and lasts approximately 2:20 minutes in performance.14 Ligeti has once again set himself an impossible task—the materials of his piece cannot suggest the piece’s conclusion within its own timeframe. This time he approaches the problem somewhat differently. Rather than trying

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to reimagine the ostinati in order to arrive at a more reasonable proportion, he focuses on resolving the tensions created by the cycle.

mm. 1-17 mm. 27-end

Section A' Section A

Ostinati RH ||: SSS LLL SSS SSS

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Example 7-5, Comparison of ostinati in A and A′ sections of 
 



The cycle contains inherent tension, one that must be released in some way in order for a cyclic piece to find closure. The source of the cycle’s tension is the overlap between the cycle’s constituent ostinati. Once the cycle has begun, it has a certain need to continue unfolding until it is complete if it is to avoid interrupting one ostinato or the other. Perhaps the most obvious method of releasing the cycle’s tension then is to let the cycle play out, to complete it. As we have seen, however, that is not always a desirable option, particularly in a piece at such a slow tempo and with relatively undifferentiated materials. Ligeti resolves the cycle’s tension in three ways: he reworks the ostinati so that they are the same length, emphasizes metric stability, and uses thematic returns. Following a contrasting B section in which only one hand strictly repeats an ostinato, the rhythmic ostinati return in A′ in somewhat altered guise. Example 7-5 shows that both ostinati are shortened. The right hand’s ostinato deletes the opening 9 attacks (SSS LLL SSS) of section A and the left changes its 4 opening attacks (note that the Section A ostinato is notated in rotation). Some of the beat lengths in A′ are also shortened. The left hand creates an apparent tempo increase without actually changing the time signature or notated tempo marking, speeding up Ss to 2 eighths and Ls to 3. Because of this change, both ostinati are shortened to the same length, 24 eighths. These shortenings result in a section in which there is essentially no cycle operating here. The ostinati do not overlap, but rather are in a metrically fixed relationship. When one restarts, so does the other, in lockstep, until the end of the piece. When the ostinati continually shift relative to one another and shift relative to the measure, new beginnings—of measures, individual ostinati, and the cycle—occur at different rates. With




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the equal ostinati, however, come fixed patterns in which there is no overlap, and beginnings of measures, ostinati, and the cycle all start together. The lack of overlap reduces the cycle’s inherent tension and need for continuity. Although these shortenings negate the cycle, a fundamental characteristic of En suspens, they do maintain a degree of continuity across the sections. Each ostinato still contains the same core series: LL SSS. This series is underlined in Example 7-5, and boxed on the score in Example 7-6, which represents the opening three alignments of the piece using its rhythmic ostinati. The series is used continually in the opening 12 measures of the piece, and in every measure of Aʹ. Despite its prevalence, however, the series remains somewhat concealed. The right- and left-hand ostinati interpret the S and L beat lengths differently throughout, but the left hand also reinterprets them for section Aʹ. In addition, the series occurs in varying places within the patterns—it is not always a beginning event, for example—and within the measure. Ultimately, this variability of rhythmic interpretation and metric placement means there is no universal version to attend to.

Example 7-6, Opening three alignments in 
  , mm. 1–12

In addition to eliminating the overlap between ostinati, Ligeti also emphasizes metric stability in A′ to resolve the cycle’s tension. He clarifies

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downbeats and strengthens metric connections between the ostinati, rather than accentuating their differences. In the A section, the perceptual strength of most downbeats was significantly weakened by ties across the bar line. In A′, however, the ties are mostly gone. They are implied, but not notated, in the right-hand ostinato in every second measure, when an L straddles the bar line. Instead, he uses clearly articulated and regularly spaced downbeats to encourage a sense of meter and stability to form. The ostinati span exactly two measures and their beginning coincides with a bar line. They sound four times in exactly the same metric positions before the piece concludes, giving listeners plenty of time to entrain and appreciate the new meter. In A′, Ligeti also introduces a constant eighth-note pulse, further strengthening a sense of meter. This addition is significant because eighths are the highest common multiple of 6/4 and 12/8 time, the notated time signatures. Running eighth notes in each hand connect seamlessly between different interpretations of beat lengths in the right and left hand, and clarify the relationship between these ostinati and the piece’s two time signatures. Although a two-against-three polyrhythm is still very strong between the right and left hand, the steady eighths emphasize the compatibility of the two ostinati. Finally, Ligeti resolves the tension of the cycle by incorporating a thematic return. In Example 7-5, the right-hand ostinati in sections A and A′ share the first five attacks: SSS LL. Ligeti emphasizes this metric connection by using the same melody in A and A′. The setting is different in many ways, not least the inclusion of running eighths just described, but accents in the upper voice make the at-pitch return abundantly clear. Similarly, comparing the left-hand ostinati in these sections, we see that the first two attack lengths are the same: LL. Ligeti also emphasizes these with an at-pitch return of the opening sonorities in section A′. The opening of sections A and Aʹ are compared in Example 7-7. When efforts aimed at increasing metric clarity are coupled with features of thematic return, they contribute to a veritable and convincing conclusion for the piece.




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Example 7-7, Thematic comparison of openings of A and A’

Ligeti’s solutions to the problems of impossibly long cycles in Désordre and En suspens indicate that rhythmic cycles can achieve resolution in many ways. These pieces are often treated as pianistic etudes, studies in which the performer can focus on some important aspect of their technique. They can also be thought of fruitfully as compositional etudes, in which the composer is focusing on different compositional possibilities. In this light, neither of these etudes are a study in how two ostinati will play out against one another. Rather, each is about allowing the composer to harness a new compositional tool. If Désordre is a study in how the process of manipulation itself can engender a sense of closure, then En suspens is a study in how cycles can participate in—without necessarily defining—a piece’s formal organization. Both are elegant solutions to the problem of a conflict between form and content.

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Notes 1

For a recent study on psychological effects of repetition, see Elizabeth Hellmuth Margulis, On Repeat: How Music Plays the Mind, (New York: Oxford, 2014). 2 Karlheinz Stockhausen explores a distinction between stopping and ending based on a piece’s approach to the arresting of sound. Pieces that stop prepare in some way for the piece’s close, while those that end do so suddenly. Similarly, pieces can begin, if they use some kind of introduction or something that prepares, or start, if they do not. In other words, piece boundaries can span a continuum of durations, depending on whether they are part of a process or a sudden event. See his The Electronic Music of Karlheinz Stockhausen: Studies on the Esthetical and Formal Problems of its First Phase. Translated by Seppo Heikinheimo. Helsinki: Suomen Musiikkitieteellinen Seura, 1972. 3 Jonathan Bernard, “Inaudible Structures, Audible Music: Ligeti’s Problem, and His Solution,” Music Analysis 6 (1987): 207–236; Jane Piper Clendinning, “The Pattern-Meccanico Compositions of György Ligeti,” Perspectives of New Music 31 (1993): 192–234; ibid., “Structural Factors in the Microcanonic Compositions of György Ligeti,” in Concert Music, Rock and Jazz since 1945: Essays and Analytical Studies, ed. Elizabeth West Marvin and Richard Hermann (New York: University of Rochester Press, 1995), 229–56; and Miguel A. Roig-Francoli, “Harmonic and Formal Processes in Ligeti's Net-Structure Compositions,” Music Theory Spectrum 17, no. 2 (1995): 242–267. 4 Richard Steinitz, György Ligeti: Music of the Imagination (London: Faber and Faber, 2003); and Denys Bouliane, “Ligeti’s six Études pour piano: The Fine Art of Composing using Cultural Referents,” trans. Anouk Lang, Theory and Practice 31 (2006): 159–207. 5 Amy Bauer, “Compositional Process and Parody in the Music of György Ligeti” (PhD diss., Yale University, 1996); and Yayoi Uno Everett, “Signification of Parody and the Grotesque in György Ligeti’s ‘Le grand macabre,’” Music Theory Spectrum 31, no. 1 (2009): 26–56. 6 Eric Drott, “The Role of Triadic Harmony in Ligeti’s Recent Music,” Music Analysis 22, no. 3 (2003): 283–314; and Clifton Callender, “Interactions of the Lamento Motif and Jazz Harmonies in György Ligeti’s Arc-en-ciel,” Intégral 21 (2007): 41–77. 7 Gretchen Horlacher, “The Rhythms of Reiteration: Formal Development in Stravinsky’s Ostinati,” Music Theory Spectrum 14, no. 2 (1992): 171–187. 8 Horlacher uses “exhausted” to refer to what I am calling “complete.” I prefer “complete” here because it offers a clear counterterm, “incomplete,” for cycles that do not go through all possible alignments. 9 Sara Bakker, “Playing with Patterns: Isorhythmic Strategies in György Ligeti’s Late Piano Works” (PhD diss., Indiana University, 2013). 10 Sara Bakker, “Incomplete Rhythmic Cycles in Ligeti’s ‘Fanfares’ and ‘Fém,’ in Ligeti’s Musical Kaleidoscopes: Essays and Analyses, ed. Jane Piper Clendinning and Clifton Callender [forthcoming from University of Rochester Press.]

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Measures per cycle: LCM (RHe, LHe) / e per measure ÆLCM (109, 144) = 15,696 / 8 = 1962 Alignments cycle: LCM (RHe, LHe) / RHe Æ LCM (109, 144) = 15,696/109 = 144 Minutes in performance: No. of measures * (tempo units per measure) / (tempo); convert remainder to base 10: * 60 / 100 Æ 1962 * 1 / 63 = 31.14 (31:08) Dividing by a fixed number of notes per measure as I have done here is somewhat theoretical, since neither hand maintains eight eighths per measure throughout the entire piece. I use it here because it provides a useful way to standardize and conceptualize the cycles for comparison purposes. Additionally, this account of the cycle is a bit of an oversimplification, because the right hand actually has two “rotations” based on its systematic process of shortening measures. Accounting for the two rotations brings the right-hand ostinato to 217e, the cycle to 3906 measures, and performance time to 62 minutes, still using 144 alignments. The way I am describing the cycle here corresponds more closely with the way I experience the alignment changes when listening. 12 Total measures: No. of e in piece / standardized measures of e Æ 1058 / 8 = 132.25 Performance: No. of measures * (tempo units per measure) / (tempo) Æ132.25 * 1 / 63 = 2.1 (2:06) 13 Performance: 75 * 6 / 90 = 5 14 Performance: 35 * 6 / 90 = 2.3 (2:20) 11

CHAPTER EIGHT IN DISGUISE: BORROWINGS IN ELLIOTT CARTER’S EARLY STRING QUARTETS LAURA EMMERY

In describing the compositional process for his string quartets, Elliott Carter explained: …I consider all these pieces an adventure. Hence, I have to do something I haven’t. I already had one adventure, and now I want another one that’s different. As a result, I think up something that intrigues me. When I’m writing, it’s not like Haydn or Mozart who wrote a whole string of string quartets one after the other. They are all more or less in the same general pattern, although they are filled with variety and differences. My quartets are in very different patterns, very different conception.1

This quote reflects Carter’s true modernist philosophy and his embrace of Ezra Pound’s slogan, “Make It New!” which, as Peter Gay observed, “modernists considered…a professional, almost a sacred obligation.”2 Indeed, Carter’s string quartets feature some of the composer’s most innovative, personalized, and boldest ideas. The first two quartets (1951 and 1959) were particularly exploratory in nature, leading to the development of techniques that mark Carter’s mature and late periods— harmonic language based on all-interval tetrachords (AITs), dense textures containing multiple polyrhythmic strands, complex counterpoint, individualization of characters, spatialization, and novel formal designs. However, a close study of the sketch material, housed at the Paul Sacher Stiftung and the Library of Congress, reveals that the works of other composers, namely Ives, Nancarrow, Bartók and Webern, served as an inspiration and even the conceptual point of Carter’s early quartets. It is quite unusual that Carter quoted and transcribed the works of others while searching for his own expression. In this essay, I examine the purpose,

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function, meaning, and different uses of existing music in Carter’s early quartets, following the typology set forth by J. Peter Burkholder in his studies on musical borrowings. Focusing on the works of Charles Ives, Burkholder develops a methodology for different procedures of using existing music. He identifies fourteen different types of musical borrowings,3 many of which can be applied to Carter’s early quartets—modeling a work or a section on an existing piece; setting an existing theme with a new accompaniment; stylistic allusion; transcribing a work for a new medium; and programmatic quotation. While the analysis of scores and sketches suggests that some borrowings in Carter are deliberate, others seem to be so well hidden and disguised that if it wasn’t for the evidence left in the sketches, they would prove very difficult or virtually impossible to detect. In his essays and interviews, Carter explained that rhythmic explorations by Stravinsky, Ives, and Nancarrow led him to realize that the boundaries of rhythmic expression could be further expanded. In his First Quartet, Carter pushed these boundaries and developed a unique rhythmic language characterized by complex polyrhythms, multiple metric layers, and a methodical shift in tempi, often referred to as “metric modulation.” Paying homage to the early forerunners of rhythmic experimentations, Carter quotes both Ives and Nancarrow in the First Quartet.4 The first movement, Fantasia, is characterized by simultaneous themes, each with a distinct speed, rhythm, and character. As a result, texture becomes stratified by means of different tempos. Fantasia contains eight principal and several subsidiary themes. The excerpt in Example 8-1, mm. 22-32, depicts the section where five themes are introduced. Theme 1 is in the second violin (m. 22), Theme 2 in the first violin (m. 22), Theme 3 in the viola (m. 25), Theme 4 in the cello (m. 27), and Theme 7 in the cello (m. 22). Theme 4, first heard in m. 27, is a direct quotation from the opening theme of Ives’s First Violin Sonata. It features all the distinguishing characteristics of quotation: it uses a pre-existent piece in a new composition; it is set apart by the prominence of the borrowing, since the borrowed theme stands out from the other voices; and it is characterized by the use of a brief excerpt.5 Carter stays true to the intervallic and rhythmic structure of the theme (Example 8-2), but transposes the pitches so that instead of starting on F3, the theme starts on the cello’s lowest pitch, C2. Hence, by being played in the cello’s lowest and richly resonant register, and marked forte and “in fuori,” the theme clearly stands out above others, suggesting that Carter wanted this quotation to be

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deliberately heard. This notion is further supported by the chart shown in Example 8-3.

Example 8-1, Elliott Carter, String Quartet No. 1, Fantasia, mm. 22-32 (a) Ives, Violin Sonata No. 1 Piano, m. 1-4

b) Carter, String Quartet No. 1 Cello, mm. 27-30

Example 8-2, Elliott Carter, String Quartet No. 1, Theme 4

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String Quartet #1 Quoted from Ives’ 2nd Violin Sonata – m. 27-29 cello alone It is heard near the beginning Theme A Ives – m. 48 B – 25 C – 70 violin D – 2nd violin m.41 22 m.

cello at 120 VII 96 VI 36 Vla 180

at climax

70 – 77

then 48 Ives

VI = 100 VII = 135 – Vla = 48 cello = 180 Viola alone – then all

Theme A Theme B

Ives theme – 27 – 30 – developed by me then 35 – A cello alone then all from 20-32 B

lyric than viola m. 70 – 77 all from

A

Ives theme combined with lyric theme m. 108 – 130

lyric theme with other theme up beat to 182 – 188 end Ives – viola – 280 – 310 – then 311 – 358 Example 8-3, Elliott Carter, String Quartet No.1, sketch of the thematic layout (transcription). (Text manuscripts, Elliott Carter Collection, Paul Sacher Stiftung, Basel)

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Carter decided on the inclusion of the Ives theme in the initial planning stages of the Quartet, as his early sketches illustrate. In the sketch transcribed in Example 8-3, Carter lays out the main thematic material for the Quartet, explaining that the first section (Fantasia) is built on a “number of musical lines and their varying combinations from the main idea in the movement,” and that one theme, which he labels Theme A, is “quoted from Ives’s Second [sic] Violin Sonata.”6 What is interesting about this sketch is that Carter initially planned to state the Ives theme by the cello alone, before he would combine it with other instruments. This would make the theme unquestionably heard and recognizable as a quotation. Further, the diagram shows how Carter planned in advance to combine Ives’s theme with other themes. Thus, he is not only paying homage to Ives by quoting his theme, but also by the manner in which he does it—by superimposing the “Ives theme” with other themes, he constructs complex rhythmic planes of polyrhythms, which is an Ivesian technique. Hence, Carter is not only using a direct quotation, but also a principle of stylistic allusion, meaning that in addition to using actual material from a pre-existent piece, this section of the Quartet also alludes to and evokes Ives’s works, styles, and textures. In addition to Ives, Carter also quotes Nancarrow’s Rhythm Study No.1 in the last movement of the Quartet, Variations. In his 1955 essay, “The Rhythmic Basis of American Music,” Carter discusses Nancarrow’s use of “unusual” polyrhythms in First Rhythm Study, which employs the combination of four distinct planes of rhythm in the piece’s “most elaborate measures” (Example 8-4).7

Example 8-4, Conlon Nancarrow, Rhythm Study No. 1, mm. 50-51

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Kyle Gann explains that this piece is an “arch-form study over ostinatos at a tempo ratio of 4:7,” within which Nancarrow superimposes twelve durations.8 Some of the durations are related to 4/4 meter at quarter note = 120 (third staff), which also marks the accents of the downbeats of the seven-note groups on the top staff, and the others to 7/8 meter at eightnote = 210 (fourth staff). Thus, Nancarrow builds multiple rhythmic planes by working primarily with duration, not tempo. In this section of the piece, the following polyrhythmic ratios emerge: Staves 1 and 2: ratio of 3:2 Staves 2 and 4: ratio of 8:3 Staves 4 and 5: ratio of 5:8

Staves 1 and 5: ratio of 5:2 Staves 2 and 5: ratio of 5:3

Nancarrow composed Rhythm Study No. 1 in 1951, which Carter himself published in New Music the same year.9 Sketches for the First String Quartet confirm that Carter was familiar with Nancarrow’s work while he was composing his First Quartet, since some of the folios in this collection indicate that he was trying to replicate a part of Nancarrow’s rhythmic design in the Quartet. For instance, the sketch in Example 8-5 shows that Carter is striving to superimpose three polyrhythms in order to create three distinct rhythmic planes. The grid on top of the page shows the general layout of ratios: 2:5:4. The excerpt is notated in the same tempo and meter as the fourth staff of the Rhythm Study No.1: 7/8 meter at tempo of eighth-note = 210. The texture and harmony on Carter’s sketch show uncanny resemblance to Nancarrow’s score: both are characterized by widely-spaced trichords. Whereas Nancarrow superimposes fifths in his trichords to obtain set (027), Carter combines fifths with minor sixths, creating set (037). This sketch does not make it into the final version of the First Quartet, suggesting that Carter was using Nancarrow’s piece as a study for developing his new technique, rather than as a direct source of material. Nonetheless, in his program notes for the First Quartet, Carter writes that “Nancarrow’s First Rhythm Study is quoted at the beginning of the Variations” movement.10 The analysis of the Quartet score shows that quotation is not literal, but rather conceptual; that is, Carter does not quote the melodies or exact rhythms, or polyrhythmic ratios and their tempos in the Variations, but the movement echoes Nancarrow. It is characterized by four instruments playing themes in distinct speeds, where the top voice plays triple-stops of widely-spaced chords, while the cello plays accented quarter-note regular downbeats at MM 120 (Example 8-6).

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Example 8-5, Elliott Carter, String Quartet No. 1: Carter’s reworking of the rhythmic ratios based on Nancarrow’s Rhythm Study No. 1 (sketch 0069v, the Library of Congress)

Example 8-6, Elliott Carter, String Quartet No.1, Variations, mm. 1-4

Perhaps the most fascinating aspect of Carter’s method of borrowing in the First Quartet is that he simultaneously uses the material from Ives and Nancarrow, both literally and conceptually. Yet, it is particularly this Quartet that defines Carter’s “mature” style. In other words, Carter created

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something quite new and identifiable by quoting an existing melody and rhythmic techniques pioneered by others decades earlier. Carter alludes to Ives in yet another way in his Second String Quartet (1959)—stylistically and programmatically. The style is Ives’s spatial music, which had intrigued Carter as early as the 1940s: the Holiday Overture (1944) and the Piano Sonata (1946) are the first examples of Carter’s exploration of what he described as “simultaneous streams of different things.”11 Carter intended to base his Second Quartet on the principle of spatiality but the task proved too difficult at this time. He sought to portray each instrument as a distinct character in operatic quartets. Hence he assigns each instrument its distinctive repertoire of intervals, speeds, colors, and gestures, which govern the tempo and texture of their individual parts, and create four character-continuities. In addition to the rigorous partitioning of the musical material among the four instruments, Carter also separates the four players in space: he indicates in the published score that the players should be seated at a greater distance from one another than usual, in order to emphasize the individualization of characters: So that contrasts of tempi and polyrhythmic textures will stand out clearly, all indications of tempi and relationships of note-values must be observed quite strictly in this work…. Within this fairly strict observance of tempi, each instrument must for the most part maintain a slightly different character of playing from the others…. To bring these differences clearly to the listener’s attention, the performers may be more widely spaced than usual on the stage so that each is definitely separated from the others in space as well as in character, although this is not necessary. 12

Carter explained that the initial idea of ascribing each instrument a specific character arose from the wish to replicate the four-part dialogue of the operatic quartets, particularly those of Verdi’s Aida and Othello.13 However, rather than evoking any operatic quartets, the Second Quartet more explicitly calls to mind Ives’s Second String Quartet (1915), which contains three movements, titled “Discussions,” “Arguments,” and “The Call of the Mountains.” Ives offered a brief program for his piece, writing on a sketch that this String Quartet is for “4 men—who converse, discuss, argue, fight, shake hands, shut up—then walk up the mountain side to view the firmament!”14 Carter’s description of the Second Quartet implies this program, particularly the first two movements, by stating that the interaction among the four characters in his Second Quartet forms three types of responsiveness: discipleship, companionship, and confrontation.15 The instruments imitate each other, cooperate with, or oppose one another.

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Hence, Carter’s Second String Quartet implements what Burkholder refers to as programmatic borrowing: it fulfills an extramusical program,16 specifically that one of Ives’s Second String Quartet. In addition to Carter’s own description of the programmatic plan for the Quartet, many sketches corroborate this notion by containing pages titled “arguments” “provocation” and “cooperation.” For instance, an early sketch for the first movement bears the inscription “provocation” (transcribed in Ex. 8-7). The first violin, taking the leading role in this movement, is meant to intimidate and provoke the other three parts with its virtuosity, bold dynamics, and bravura character.

Example 8-7, Elliott Carter, String Quartet No.2: Allegro fantastico, “Provocation”

It is not a secret that Carter admired Ives and Nancarrow, and that both composers profoundly affected his own compositional development. Therefore, it is perhaps not surprising that certain musical ideas—such as quoting existing themes of Ives, replicating Nancarrow’s polyrhythmic layers, implementing spatialization, or even borrowing Ives’s programmatic ideas—appear in Carter’s own works. However, what is surprising is that sketches for his Second String Quartet reveal that Carter borrowed ideas from composers he never publicly recognized as his influences—Bartók and Webern. Although he admired the two, especially Webern, he never mentioned that their works, or any of their ideas, served as any basis or inspiration for his music. Carter’s quartets are often compared to Bartók’s for their importance,17 virtuosity, introspection, structural rigor, and emotional intensity. For instance, David Schiff observes that Carter subliminally identified the synthesis of the European and American modernism in his First Quartet

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with stylistic allusion that echoes Bartók’s Fourth String Quartet.18 Sketches show that while the connection of Carter’s First String Quartet to Bartók’s Fourth is suggestive, the relationship between Carter’s Second and Bartók’s Fourth is quite certain. Bartók’s Quartets Nos. 3 (1927), 4 (1928) and 5 (1934) are intensely chromatic atonal works, characterized by short motives of about three or four notes in length.19 Motives usually outline a span of a perfect fourth, which is then chromatically filled. For instance, as Joseph Straus observes, in the first movement of the Third Quartet, the motive of a perfect fourth is subdivided into a major second and minor third. The motive of the Fourth Quartet is more chromatic, containing three minor seconds within a minor third (Example 8-8):20 (a) String Quartet No. 3, Prima Parte, mm. 87-89

(b) String Quartet No, 4, I. Allegro, mm. 1-2

Example 8-8, Béla Bartók, String Quartets Nos. 3 and 4: motivic characteristics

Carter shows his familiarity with Bartók’s treatment of motivic ideas, as he inscribes on one of the folios pertaining to the fourth movement of the Second Quartet, “for Bartók” (Ex. 8-9). The segment features a staggered entrance of voices: a sustained Eb in the viola; an outline of a minor third, D-F, in the first violin; a C# in the cello; and a block third, CE, in the second violin. Unquestionably, Carter follows Bartók’s method, showing that he is outlining a span of a perfect fourth, which he fills-in chromatically with the violins playing thirds and the lower strings a major

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second. Although this measure does not literally translate into the published score, it is quite significant because it suggests that Carter might have borrowed Bartók’s technique, which prevails in all of his music since the First Quartet: dissonant and atonal compositions, with densely contrapuntal lines, and staggered entrance of voices, each introducing pitches that chromatically fill in smaller intervals (thirds or fourths). To this, Carter adds new elements—he combines the pitches to obtain the allinterval tetrachords (0146) and (0137). These constitute the fundamental harmonic design of this Quartet, which he combines with other tetrachords to complete the aggregate.

Example 8-9, Elliott Carter, String Quartet No. 2: “For Bartók,” transcription (Elliott Carter Sammlung, Paul Sacher Stiftung)

Through numerous revisions of the segment in Example 8-9, Carter develops m. 610, shown in Example 8-10. This measure truly exemplifies how Carter merged Bartók’s motivic treatment with his own. Reminiscent of Bartók’s style, the cello outlines a perfect fourth (E-A), which is chromatically filled in by the pitches in the viola (G-Ab) and the second violin (Gb-F). The remaining pitches chromatically complete a span of another perfect fourth, (Bb-Eb). Further, the individual instrumental lines feature the intervals of thirds (the violins), seconds (the viola), and fourths (the cello). Carter carefully chooses the order in which pitches are introduced, so that his harmonic language, based on AITs, upholds. Hence, this measures features both forms of AITs—(0146), occurring between the second violin and the cello twice, and (0137), between the second violin and the viola. The first violin introduces the remaining pitches needed to complete the aggregate. This measure is just one instance where Carter effectively blends two methods—his own and a borrowed one. The sketch leading to it (“for Bartók”) is a direct link to Bartók, and shows how Carter took another composer’s conceptual

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method and combined it with his harmonic language to derive an identifiably unique harmonic language.

)

)

)

Example 8-10, Elliott Carter, String Quartet No. 2, IV-Allegro, m. 610

The sketches for the Second Quartet reveal even more curious borrowings. Found among the folios in this collection is a sketch titled “Webern” (Example 8-11). This sketch, in which Carter re-composes Webern’s Six Bagatelles for String Quartet, No. 6, Op. 9, is extraordinary for several reasons: it is the only known example in which Carter transcribes a piece of another composer; it shows a close connection to Webern; and it may be the only direct indication of Carter using a work of another composer as a starting point of his own. Although Webern’s Bagatelle uses four staves, Carter reduces the parts to two, adding a third staff on the second system, in m. 5. The transcription abruptly stops on the downbeat of m. 6, leaving a large portion of the page blank. This layout of condensing the staves suggests that Carter intended to at least finish the transcription of the entire piece— the piece is only nine measures in length. Yet, the excerpt stops about halfway through. Thus, the questions remain—why did Carter transcribe this Bagatelle, and how did he intend to use it in his Second String Quartet?

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(a) Carter’s transcription of Webern’s Bagatelle No. 9 (Elliott Carter Sammlung, Paul Sacher Stiftung)

Example 8-11, Borrowings from Webern

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(b) Webern’s Bagatelle No. 9, transcription of Carter’s transcription

(c) Anton Webern, Bagatelles for String Quartet, No. 6, Op. 9, mm 1-6

Example 8-11, Borrowings from Webern, cont’d

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Although reduced and compressed, Carter’s transcription stays true to the original—tempo is marked at MM 84 in 3/4 meter, and the pitches, all written in their sounding registers, rhythm, articulation, and most of the expressive and tempo markings are written out in detail. Unlike the “for Bartók” sketch, which Carter wrote as a dedication to the composer he admired, the Webern sketch does not bear characteristics of a dedication, but rather of a study. This sketch is found among the folios in which Carter works out the Introduction of the Quartet, therefore making it one of the earliest sketches for the Second Quartet. Most of these early sketches pertain to the harmonic ideas and show that Carter greatly struggled while developing his harmonic language. Dozens of pages contain systematic combinations, transpositions, inversions and other transformations of two-, three-, four-, five-, six-, eight-, and twelve-note chords.21 The sketches also indicate that Carter was trying out several different methods, including serialism, while searching for harmonic grammar that would be suitable for his expression. He admits to having serious difficulties in a letter to Goffredo Petrassi, in which he writes: I have just finished the Second String Quartet, which has caused me much work, much perplexity. I had certain ideas for my piece, which my musical technique did not allow me to develop, or help me find other things that would work with the ideas which I began. Even serialization did not help me, even though I tried it several times.22

Confirming the conceptual difficulties in the letter, many pages of preliminary harmonic sketches show Carter working with twelve-tone rows, exploring their properties via segmentation into trichords, tetrachords, and hexachords, and their many transformations. It is these “row” sketches that truly reveal Carter’s frustration while composing the Second Quartet, since he attempted using the method here even though it is well known that he strongly disliked it.23 Thus, it becomes not too surprising that Carter turned to Webern’s Bagatelle in his search of harmonic language that incorporates the twelvenote aggregate in a non-serial manner. Written in 1913, Bagatelle No. 6 is an example of Webern’s pre-serial, “free-atonal” composition. Nonetheless, Webern displays a specific approach to using the twelve-note aggregate, which involves a consistent deployment of chromatic segments.24 Every note is heard clearly, with a distinct color, expressivity, or gesture. This piece greatly appealed to Carter for several reasons: the piece opens with a (0137) AIT, which is one of the structural harmonic blocks of the Second Quartet; by the end of the first measure, all interval

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classes have been introduced both vertically and linearly; and by beat two of the following measure, the aggregate is complete. Both Webern’s control of pitches and his treatment of intervals are of particular interest to Carter—the Second Quartet not only incorporates the combination of tetrachords (the two forms of AITs with the “left over” tetrachords) to complete the aggregate, but each instrument also works with a distinct repertoire of intervals to emphasize their individuality.25 Further, the fragmented, sparse texture of the Bagatelle with a variety of timbres, gestures, expressions, and articulations are all characteristics of the Quartet’s Introduction (Example 8-12). Sketches and correspondence from the Elliott Carter Collection show us that for Carter, developing his own expression was not an easy task, especially for a composer who sought to create a “new adventure” with each piece, to say something he has not said already, and to “make it new.” But it is an idea worth noting that a composer who took Ezra Pound’s words to heart deliberately used the materials and ideas from others to create that something new. While Carter’s quartets represent his most original compositions and feature Carterian expression, many of his ideas in the early quartets derived from the techniques of his contemporaries: for his rhythmic explorations in the First Quartet, he turned to Ives and Nancarrow, whom he both quotes and alludes to in the Fantasia and Variations movements, respectively. But, according to Carter, they have only scratched the surface of rhythmic expression; their concept of rhythmic exploration was concerned with local detail, while Carter sought to elevate it to the large-scale structural level, just as twelvetone compositions governed both the small and large-scale structures of compositions. Carter achieved this goal of rhythmic expression in his compositions of the 1980s, all of which are structurally based on longrange polyrhythms—rhythms that guide both the large-scale and local rhythmic design of a composition. For his experimentation with the spatial element in his Second Quartet, Carter was undoubtedly influenced by Ives. The concept of spatialization remained an important effect throughout Carter’s oeuvre: Double Concerto (1961), Concerto for Orchestra (1969), A Symphony of Three Orchestras (1976), or his later concertos (2000 ASKO Concerto and 2002 Boston Concerto 2002), in which Carter continues to separate the players into distinct groups both spatially and musically. Lastly, while developing his own harmonic language, Carter looked for answers in Bartók and Webern, examining their methods of completing the aggregate without serialization, and typically working with smaller intervals or segments, and then systematically filling them in chromatically. Carter continued to work with this idea from the First

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Quartet on, until he developed his own harmonic grammar, consisting of superimposed all-interval tetrachords in the 1950s and 1960s, and alltrichord hexahords and all-interval twelve-note chords from the Third Quartet on (1971).

(0146)

(0146)

(0146)

(0146)

(0146)

(0146)

(0146)

Example 8-12, Elliott Carter, String Quartet No. 2, Introduction

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The idea of musical borrowings is an old one, as David Metzer notes— the practice can be traced to the use of chants as tenors in Notre Dame organum, and ever since, composers have continued the practice of borrowing pre-existent material. What is surprising in the case of Carter’s borrowings is that he never spoke of Bartók and Webern as influencing his compositional development. Yet, the sketches clearly show that they did. The “For Bartók” sketch suggests that Carter perhaps intended to pay tribute to the composer, in the same way he symbolically quoted Ives and Nancarrow in the First Quartet. But neither a direct quotation from any of Bartók’s quartets nor a distinct paraphrase, resembling the aforementioned sketch, makes it to the final round. Similarly with the Webern sketch, which is not an example of dedication to a composer he admired, but of one composer studying the method of another, Carter does not incorporate the excerpt of the Bagatelle directly into his Quartet. Rather, these borrowings are disguised: Carter examines the composers’ harmonic language, method of fragmentation, expression and tone color, which he combines with his own to generate a unique expression. With his methods of borrowing in early quartets, Carter effectively tied together different stylistic and cultural strands, and showed new approaches to borrowing by combining different practices, including quotation, allusion, modeling, and paraphrase.

Notes 1

Laura Emmery, “An American Modernist: Teatime with Elliot Carter,” Tempo 67/264 (Apr. 2013): 22-29. 2 Peter Gay, Modernism: The Lure of Heresy: From Baudelaire to Beckett and Beyond (New York: Norton, 2008), pp. 46, 106. 3 In his study on musical borrowings in the music of Charles Ives, J. Peter Burkholder identifies fourteen different ways Ives incorporated existing musical material into his compositions: 1. Modeling a work or a section on an existing piece; 2. Variations on a given tune; 3. Paraphrasing an existing tune to form a new melody, theme, or motive; 4. Setting an existing tune with a new accompaniment; 5. Cantus firmus; 6. Medley, stating two or more existing tunes; 7. Quodlibet, combining two or more existing tunes or fragments in quick succession; 8. Stylistic allusion, alluding not to a specific work but to a general style or type of music; 9. Transcribing a work for a new medium; 10. Programmatic quotation; 11. Cumulative setting, in which the theme is presented complete only near the end of a piece; 12. Collage, in which a swirl of quoted and paraphrased tunes is added to a musical structure; 13. Patchwork, in which fragments of two or more tunes are stitched together; and 14. Extended paraphrase. See Burkholder, All Made of

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Tunes: Charles Ives and the Uses of Musical Borrowing (New Haven: Yale University Press, 1995), pp. 3-4. 4 Elliott Carter, “String Quartets Nos. 1, 1951, and 2, 1959 (1970),” in Collected Essays and Lectures, 1937-1995, ed. Jonathan W. Bernard (Rochester: University of Rochester Press, 1997), p. 233: “This quartet, for instance, quotes the opening theme of Ives’s first Violin Sonata, first played by the cello in its lowest register after each of the other instruments has come in near the beginning. A rhythmic idea from Conlon Nancarrow’s First Rhythm Study is quoted at the beginning of the Variations. These two composers, both through their music and their conversation, had been a great help to me in imaging this work and were quoted in homage.” See also Carter, “The Rhythmic Basis of American Music,” in Collected Essays and Lectures, p. 61-62, for his discussion of Nancarrow’s polyrhythms in the First Rhythm Study, which employs the combination of four distinct planes of rhythm in the piece’s most elaborate measures (mm. 50-51). 5 David Metzer, Quotation and Cultural Meaning in Twentieth-Century Music (Cambridge: Cambridge University Press, 2003), p. 4. 6 See text manuscripts, Elliott Carter Collection at the Paul Sacher Stiftung (Basel, Switzerland). Here Carter mistakenly cites that the theme is quoted from Ives’s Second Violin Sonata instead of the First Violin Sonata. 7 Carter, “The Rhythmic Basis of American Music,” in Collected Essays and Lectures, p. 61-62. 8 Kyle Gann, The Music of Conlon Nancarrow (Cambridge: Cambridge University Press, 1995), p. 70. 9 David Schiff, The Music of Elliott Carter, 2nd ed. (Ithaca: Cornell University Press, 1998), p. 31. 10 Carter, “String Quartets Nos. 1, 1951, and 2, 1959” (1970), p. 233. 11 See Jonathan Bernard, “The True Significance of Carter’s Early Music,” in Elliott Carter Studies, eds. Marguerite Boland and John Link (Cambridge: Cambridge University Press, 2012), pp. 3-32. Bernard observes that the first movement of the Piano Sonata exhibits such abrupt shifts in music characters, that rather than seeing them alternate, they overlap, usually each having its distinct speed (pp. 15-16). In his discussion of the Holiday Overture, Bernard points to the literal projection of contrasting rhythmic strata in the piece, most notably in the section starting in m. 103 (pp. 18-20). Bernard cites Carter revealing that the Holiday Overture was one of his first works in which he began applying the idea of “simultaneous streams of different things going on together” (Bernard, p. 18; also Allen Edwards, Flawed Words and Stubborn Sounds: A Conversation with Elliott Carter [New York: Norton, 1971], p. 101). 12 Carter, “String Quartet No. 2: Performance Notes,” in Elliott Carter: The String Quartets [score] (New York: Associated Music Publishers, Inc., and Hendon Music, Inc., a Boosey & Hawkes Company, 1998), pp. 120-121. 13 Schiff, The Music of Elliott Carter, p. 73. 14 Burkholder, All Made of Tunes, p. 348.

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Carter, “String Quartets Nos. 1, 1951, and 2, 1959” (1970), p. 234. Burkholder, All Made of Tunes, p. 4. 17 Jonathan Bernard, “The String Quartets of Elliott Carter,” in Intimate Voices: The Twentieth-Century String Quartet, Vol. 2, ed. Evan Jones (Rochester: University of Rochester Press, 2009), pp. 238-275. Bernard notes, “[I]n terms of chronological distribution and degree of importance among the composer’s works in general, it would be difficult to think of any group of string quartets of the post1945 era more like Bartók’s than those of Elliott Carter” (p. 239). 18 Schiff, The Music of Elliott Carter, p. 23. 19 Joseph Straus, “The Pitch Language of the Bartók Quartets,” in Intimate Voices: The Twentieth-Century String Quartet, Vol. 1, ed. Evan Jones (Rochester: University of Rochester Press, 2009), 70-111: pp. 70-71. 20 Ibid., pp. 71-72. 21 Eventually, this methodical formulation will result in his Harmony Book. In my dissertation, I argue that it was precisely during the process of composing the Second String Quartet (1958-1959) that Carter formulated his Harmony Book. Unlike sketches of other pre-Harmony-Book-compositions, the Second Quartet sketch collection contains dozens of pages where Carter systematically explores all combinations and transformations of dyads and chords. See Chapter 2 in Emmery, “Evolution and Process in Elliott Carter’s String Quartets.” Ph.D. diss., University of California, Santa Barbara, 2014. 22 This is my translation of a letter Carter wrote to Goffredo Petrassi on May 11, 1959. The original letter, written in French, is printed in Felix Meyer and Anne C. Shreffler’s Elliott Carter: A Centennial Portrait in Letters and Documents (Suffolk: The Boydell Press, 2008), p. 158: “J’ai presque fini un deuxième quatuor à cordes qui m’a coûté beaucoup de travail, de perplexité. Toujours j’ai des idées pour des moments ou des endroits dans une composition et ma technique musicale ne m’aide pas à les développer ou même à trouver d’autres choses qui vont avec les idées avec lesquelles j’ai commencé. Même la sérialisation ne m’aide pas— quoique je l’ai essayée plusieurs fois.” 23 In his 1960 article, “Shop Talk by an American Composer (1960),” in Collected Essays and Lectures, 1937-1995, ed. Jonathan W. Bernard (Rochester: University of Rochester Press, 1997), 214-24, Carter says that he never used the twelve-tone system in his music, because he found it inapplicable to what he was trying to do. As such, he found it to be more of a hindrance than help. However, he does state that he was familiar with the method and that he studied the important twelve-tone works, many of which he admired, “out of interest and out of professional responsibility” (pp. 219-220). 24 For a detailed discussion on the harmonic structure of Webern’s Bagatelle No. 6, see Benjamin Davies, “The Structuring of Tonal Space in Webern’s Six Bagatelles for String Quartet Op. 9,” in Music Analysis 26/1-2 (2007): 25-58; Sallmen, “Motives and Motivic Paths in Anton Webern’s Six Bagatelles for String Quartet, Op. 9,” in Theory and Practice 28 (2003): 29-52; and Chrisman, “Anton Webern’s 16

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‘Six Bagatelles for String Quartet,’ Op. 9: The Unfolding of Intervallic Successions,” in Journal of Music Theory 23/1 (Spring 1979): 81-122. 25 In the Second Quartet, Carter assigns each instrument its own repertoire of intervals, to emphasize their individuality. The first violin uses the intervals of a minor third and a perfect fifth; the second violin is assigned a major third, major sixth, and a major seventh; the viola is characterized by the intervals of a tritone and a minor seventh; and the cello uses a perfect fourth and a minor sixth.

CHAPTER NINE COMPOSITIONAL SPACES IN MARIO DAVIDOVSKY’S QUARTETTOS INÉS THIEBAUT1

Mario Davidovsky is best known as the composer of Synchronism(s), pieces that involve electronic sounds. The fame of these pieces has tended to obscure a basic fact about the composer: he has written more acoustic works than electroacoustic, and cares as deeply about pitch as he does about gesture, temporality and timbre. The pitch organization in his music has not been well understood: the music is in fact aggregate-based, but not serial. This is an important distinction, for I argue that the music is nonetheless based on a principle of ordering, an ordering based on a cyclical conception of the aggregate, rather than a series. What is even more interesting, this type of pitch organization is inherently connected to Davidovsky’s musical flow. As we will see in the following pages, pitch organization fuels musical motion by three principal means: by maintaining, establishing or reestablishing symmetry, by unfolding interval cycles at different levels of structure, and by aggregate completion. Often, these three sources of musical energy are mutually reinforcing, but not always. The concepts of expectancy, resolution and realization will thus become important when interpreting this music. In the following pages we will see instances of expectancy (symmetrical and cyclical) being fulfilled (i.e., when a pc is needed to complete both an unfolding cycle and the aggregate, and this pc is indeed present in the music), as well as thwarted resolutions (i.e., when a different musical event is found instead of the expected one). *** Figure 9-1 shows the familiar interval class 5-cycle in Mod 12. I’ve chosen this traditional tonal space for several reasons. First and foremost,

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Davidovsky’s musical surfaces in the Quartettos often outline the fifth as an essential interval, both vertically and linearly.2 The “fifthyness” of his music is a quality that we cannot ignore, nor assume to be isolated to a particular piece.

Figure 9-1, the 5-cycle

The second reason is that his large-scale formal structures are often also governed by interval cycles (mostly the 5- and 3-cycles), as we will see. The third reason is that this space is particularly well suited to show symmetrical collections, for I will also show how Davidovsky tends to establish and later challenge symmetrical axes as the music progresses in both complexity and time. These axes are often singled out, not just by their particular salient presence in a musical passage, but also by their absence when they are expected. The fourth reason is Davidovsky’s interest in the aggregate, which makes this 5-cycle space ideal. The only other cycle besides the 5-cycle that goes through the 12 pitch classes before repeating is, as we know, the chromatic cycle. Let us explore this last reason more deeply. Figure 9-2 shows the 1-cycle embedded within the 5-cycle. These two cycles have a special relationship: all you need to do to transform one into another is multiply each of the elements by 5 (e.g., 5 x 5 = 25, in MOD 12 = 1). This relationship, called M5 and explored already by many composers and theorists,3 creates several useful compositional possibilities in our space: first, once embedded, both cycles preserve the position of the 3-cycle, and thus allowing it to maintain a pivotal quality between the two. Second, they both also share the possibility of mapping the 2-cycles (the even, and the odd) by skipping every other pitch-class within the space

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(the difference being that the 2-cycle is descending (i.e., clockwise) through the 5-cycle, while it is ascending (i.e., counterclockwise) through the 1-cycle).

Figure 9-2, the 1-cycle embedded within the 5-cycle

Figure 9-3a, the 3-cycle preserved in the space

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Figure 9-3b, the 2-cycle descending and ascending through the space (only the even 2-cycle is shown)

This mapping is, of course, also true for the 4-cycle (we would have to skip every three pitch-classes within the space) and the 6-cycle. In other words, both 5- and 1-cycles share their ability to not just map their own cycles (that is, the aggregate), but all the other ones as well. *** Figure 9-4a shows the first 4 measures of Quartetto No .4 (written in 2005, for clarinet and string trio). Figure 9-4b maps the starting chord within both the 5- and 1-cycles, which consists of three fifth-related pitches (B, C#, F#) and the more lopsided E#. The figure also shows the potentiality for symmetry, which is happily confirmed once the singled-out G harmonic enters in the cello at m. 4. Figure 9-4c maps the resulting symmetrical pentachord. The symmetry (in both of our cycles) is thus established around pc0. We will call this SymI0.

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1

Figure 9-4a, Quartetto No. 4, mm. 1-4; an asymmetrical chord made symmetrical by the entrance of G at the end of m. 4. © Copyright by Edition Peters, London. Reproduced by kind permission of the publishers.

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Figure 9-4b, Mapping of the initial chord within the 5- and 1-cycles, and potential symmetry completion

Figure 9-4c, Symmetry completion once pc7 enters at m. 4

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Figure 9-5a shows mm. 5-8, in which three events occur: the maintenance of SymI0 (m. 5), introduction of SymI1 (m. 6), and aggregate completion (m. 8). The next two notes we hear, the pizzicato Ab in the viola and the E on the violin at m. 5 clarify SymI0.

Figure 9-5a, three events follow: maintenance of SymI0 (m. 5), introduction of SymI1 (m. 6), and aggregate completion (m. 8)

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Figure 9-5b, pc4 and pc8 at m. 5 reinforce the initial symmetry around pc0

Yet SymI0 is short-lived. The viola and violin continue to play a combined gesture through m. 6 that introduces four new pitches (C, Eb, Bb, Db) that contain their own axis of symmetry around the Bb/Eb dyad within the 5-cycle, and around the C/Db dyad within the 1-cycle. We will call this SymI1.

Figure 9-5c, introduction of SymI1 in mm. 5-6

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The compositional decision to introduce different symmetrical collections (both linearly and vertically) as the music progresses is a technique often found in Davidovsky’s Quartettos. An attentive reader might have noticed that there is only a fifth-related pair left to complete the aggregate (A and D), which shows the intentionality of the two collections presented in the music thus far. Indeed, A is the next pitch on both the viola and the cello at the end of m. 6. This A has been singled out, for it is the first unison of the work. The remaining D is played tenuto in the very high register of the cello on m. 8, and has a clear double function: to complete the first aggregate, and to start a new one as the music moves into a new section. This D in fact does not occur alone. Figures 9-6a and 9-6b show the complete sonority at m. 8 with its potential symmetry and actual realization. This chord is a transposition at T8 of our initial (0157) tetrachord. If somehow the listener was able to retain in her short-term memory how the original chord proceeded, or somehow felt SymI0 when the cello’s G sounded in m. 4, she might consider the potentiality of this transposed sonority in m. 8. The expected symmetry would now be around pc2, and an Eb would be expected to complete the motion. In the context of voice leading, this would have been relatively easy (move the cello’s D up a half-step to Eb). Yet instead the D moves down to C, breaking any possible expectation and moving the music forward into new territory. We would call this unfulfilled symmetry around pc2, SymI4.

Figure 9-6a, (0157) at T8 not fulfilling the expected SymI4 as the cello moves to C instead of Eb

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Figure 9-6b, mapping of (0157) at T8 in m. 8 within the 1- and 5-cycles; Pc0 breaks any possible expectation of symmetry completion

Figures 9-7a and 9-7b show the solo clarinet line in mm. 13-21, when it is heard for the first time in the piece. The first three pitches follow a clear 5-cycle motion (Ab, Eb, Bb), and the next note, a low D (m. 18), completes yet again another transposition of our opening (0157) tetrachord, this time, at T9. Once again, we might expect a symmetrical fulfillment, now waiting for a possible E. This potential symmetry would thus be expected around pc3. We will call it SymI6. At m. 19 the low D leaps a fifth to A (which magnifies a possible Eb/A axis fulfillment), yet the expectation is once again broken when we hear C instead of E in m. 20, the same pitch class that broke the previous SymI4’s expectations. This C is marked piu piano, as if it were shy, knowing that maybe it shouldn’t be there.

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Figure 9-7a, clarinet solo line mm. 13-21; (0157) at T9 not fulfilling the expected SymI6 as the clarinet moves to C instead of E

Figure 9-7b, mapping of (0157) at T9 in mm. 13-21; Pc0 once again breaks the possible expectation for symmetry completion

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There are multiple examples of symmetrical unfulfillment throughout Quartetto No. 4. The expectation for unsymmetrical collections to resolve into symmetrical collections as the music progresses from one point to the next is established right from the start as a particular type of musical energy. We’ve seen thus far three consecutive reiterations of the opening asymmetrical (0157) tetrachord, the second at T8 and third at T9. Only the initial (0157) was able to resolve into the symmetrical pentachord (01268) as Davidovsky completed the first aggregate. What is intriguing about these first 21 measures is the constant upward shift in the symmetrical axes implied, both fulfilled and unfulfilled. SymI0 was our first fulfilled symmetry. On the musical surface this fulfillment meant that pc7 was indeed heard, and more than that, it was heard alone, as if the pitch itself were conscious of its role. SymI4 was implied around pc2, but not fulfilled due to the intrusion of pc0 instead of pc3. Davidovsky could have played out SymI4’s longing for the Eb in the clarinet’s solo line that immediately follows. SymI6 was indeed implied around pc3, yet it was also disrupted by the presence of pc0. SymI1, fulfilled around pc1, was found in mm. 5-6 during the brief presence of the (0135) tetrachord needed in order to complete the first aggregate. As we’ve seen, the rising chromatic axes (fulfilled or unfulfilled) remain mostly abstract from the surface of the music except for very specific moments: the initial pc0 axis (its presence, its weight) could lay behind the stubborn C that keeps breaking the symmetrical expectations of SymI4 and SymI6, and pitch D was the last pc to complete the first aggregate and served as transition into the potential SymI4. *** Let us now observe another example in which aggregate completion and cyclical presentation of symmetrical collections converge on the musical surface. Figures 9-8a through 9-8e show the opening 9 measures of Quartetto No. 3 (written in 2000, for piano and string trio). The initial tremolo A-C dyad in the piano’s right hand is transferred to the cello and viola on the last beat of m. 4 (heard as harmonics, extending the ic3 sonority). At m. 4, the piano’s right hand moves to a second ic3 dyad (DB). So far, the resulting (0235) sonority establishes symmetry around the A/D dyad. We will call this symmetry SymI11.

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Figure 9-8a, Quartetto No. 3, mm. 1-9; completion of the first aggregate

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Figure 9-8b, mapping of (0235) within the 5-cycle; the ic3 dyads heard in mm. 1-4 between the piano’s right hand, cello and viola establish SymI11

The piano’s left hand and violin have other plans, however: the piano’s left hand ic5 dyad C#-G# in mm. 4-5 is answered by the violin’s F#-B i5 dyad on m. 6. These two ic5 pairs (placed carefully on opposite sides of the range spectrum) have their axis of symmetry around the C#/F# dyad. We will call this SymI7.

Figure 9-8c, mapping of (0123) and added to the previous (0235); the two ic5 dyads heard in mm. 5-6 between the piano’s left hand and the violin revolve around SymI7 (pc11 is doubled in both collections)

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Two new dyads appear in mm. 7-8. A new minor third dyad (G-Bb) in the piano is presented now in an extreme range combining both hands. The violin’s F-Eb tremolo imitates the piano’s opening gesture, and it gains momentum as it speeds up over the A-C minor third in the rest of the strings (our original ic3 pair, which is still sustaining). This F-Eb dyad stands out by the fact that it is the only ic2 dyad in the excerpt: it fills the space and prepares the last pc needed to complete the aggregate. Indeed, this tremolo gesture gravitates towards E, which is introduced at m. 9. Once again this last note is singled out as we saw in Quartetto No. 4, only this time with the peculiar color of the piano string pluck.

Figure 9-8d, completion of the aggregate; the ic2 dyad in the violin gravitates towards the Bb-E axis, as the piano’s final ic3 dyad is heard in a quasi-cadential motion in m.7 (G-Bb); the final pc4 is heard in m. 9

The curved arrow from B to E in figure 9-8d signals the possible fifthrelation operating deep within the passage. The B was, indeed, the only pitch doubled in the two initial collections presented, as it was part of both a ic3 dyad, and a ic5 dyad. In the music we find that the piano’s right hand B is actually sustained for three full measures. The quasi-cadential final ic3 dyad in the piano is written out as voices 2 and 3, keeping B sustained through the measure. As in Quartetto No. 4, the final pitch that completes the aggregate (here, the E) does not occur alone, as the music transitions into a new section.

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In regards to the symmetrical implications of the passage, it is interesting to consider Figure 9-8e. The appearance of the final ic3 G-Bb dyad as the quasi-cadential motion in m. 7 could not only signal the upcoming completion of the aggregate, but also “resolve” both the SymI11 and SymI7 into SymI9. In the figure, we’re taking into consideration the ic3 dyads only for the establishment of this possible SymI9. The axis of SymI9 around a D/G dyad within our 5-cycle would not correlate with the musical surface’s pull towards pc4, which as we saw completed the aggregate in m. 9. The dichotomy between symmetrical axis and pitch centricity (if we are able comfortably to call these targeted, temporary pitches as such) could very well be a topic for an entire new article, and much too complex to be dealt with here. For now, let us state that the axes can manifest themselves, as we saw in our previous examples, both within deeper levels of structure and surface realizations, as they fuel further musical motion. Yet, at a surface level, temporary targeted pitches other than these axes can in fact govern salient features of the music via cyclical relationships and aggregate completions.

Figure 9-8e, possible resolution of SymI11 and SymI7 into SymI9 once we take into consideration the final ic3 G-Bb dyad in context of the previous ic3 dyads

So far we’ve been discussing examples in which the aggregate and cyclical intervallic relationships are very close to the surface of the music. Yet this is not always so. I will focus the remaining portion of the article on Quartetto (the first of the four, written in 1987, for flute and string trio), and show the more structural pitch patterns that govern this piece. Figures 9-9a through 9-9c will map the first 20 measures of Quartetto.

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Figure 9-9a, the first 20 measures of Quartetto

The piece starts with a string unison pitch C marked triple piano, with no vibrato. Incidentally, the triple piano will remain as the overall dynamic

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until m. 40; I will be talking about that moment later on. If we pay attention to the emphasized pitches that follow in the flute’s phrase (an emphasis based both on rhythm and contour), we can hear the F on m. 4 as a response to this opening C, following a low Eb and the high Bb in mm. 5-6 as responses to this first 5-cycle initiation. The strings then present yet another unison, this time on Ab in m. 10, moving along our 5-cycle quite nicely; and finally, after the lingering Ab and Bb, the flute rests on the Db in mm. 13-14. The quarter note rest that follows provides a small moment of contemplation. Symmetry is established around the flute’s low Eb and high Bb dyad, and maps the pitches that started and ended this 5-cycle gesture (C and Db) onto one another. We will call this symmetry around the Bb-Eb dyad SymI1.

Figure 9-9b, the initial flute gesture of Quartetto, mapped within the 5-cycle. SymI1 is established around pc10/3, which maps the initial unison C into the final Db in the flute

Three main events follow: the unison line in the viola, violin and flute on mm. 17-18 first rests on a diminished triad on m. 19, and then moves to a full-ensemble unison D on m. 20. The original string unison on C shifts its power towards this strong full ensemble unison on D, and SymI1 is challenged. I will comment on the diminished triad later, for now let us stay focused on the motion of the structural pitches from C to D in these measures.

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Figure 9-9b, the structural unisons on pc1 and pc2 within the overall 5-cycle gesture of the first 20 measures; the entrance of pc2 in m. 20 challenges SymI1

Figures 9-10a and 9-10b follow the events that outline the structural arrivals from mm. 22-36, the section that follows this long, unison D in m. 20. The unison D triggers its lower 5-cycle companion (A) in m. 22. This ic5 dyad will govern the six following measures, and it is the cello’s double stop in m. 28. A new ic5 pair follows (E-B) in mm. 31-32, with the B emphasized as a full-ensemble unison. If we were to expect a full mirrored version of our opening gesture, the C#-F# ic5 pair would follow, and a new symmetry would be expected around the E/B dyad. Davidovsky does provide this last ic5 pair, although not clearly, in mm. 34-35. In fact, it is so faint it almost seems unfulfilled purposely, as if the C# is not supposed to have a full presence here. We will call this new symmetry, SymI3, and note that it also maps the starting pitch D into the faint C#, just as SymI1 mapped the starting C into Db in mm. 1-15.

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Figure 9-10a, Quartetto, mm. 22-36; the opening ic5 cycle gesture mirrored now from D to C#: ic5 pair D-A moves to ic5 pair E-B to faintly reach the final ic5 pair F# - C#

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Figure 9-10b, mapping of the structural pitches in mm. 22-36, from the strong unison pc2 in m. 20 to the faint pc1 in m. 36; SymI3 is established around the E-B dyad, mapping D onto C#

Let us follow this thought for a moment: Figure 9-11 combines the initial 5-cycle from C to Db as it was unfolding in the first 15 measures of the piece and its mirrored response, the 5-cycle from D to the not so clear C#.

Figure 9-11, combined mapping of two gestures: the initial pc0-pc1 from mm. 115; and the inverted response pc2-pc1 from mm. 20-36

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Both gestures move towards pc1 (spelled Db in m. 20, then C# in m. 36) yet it is the second gesture from D to C# that feels incomplete, as the C# is realized very weakly within the surface of the music. Both gestures combined in our space show the gravitational pull towards pc1, and also show the absence of pc7 as a structural tone, which would complete the aggregate at this deeper level. The piece in fact ends with a full-ensemble union C#. This last C# is also marked triple piano, without vibrato, just like the first C of the piece, and it’s the only full-ensemble C# unison in the entire work (in fact, there are only six full-ensemble unisons, and as we will see shortly, and four of them occur in the first section of the piece). It could be that the D’s hunt for C# is one of the narratives for the work as a whole. The absence of pc7 as a structural tone in these initial 36 measures could also very much be part of this narrative, as it is the fifth related pair to pc0. This narrative, I fear, is too long for the scope of this paper. I will, instead, take a moment to clarify the most salient non-unison simultaneities that occur in these measures, before investigating the more interesting unison simultaneities that seem so structural to the work. Figure 9-12 shows the diminished trichord previous to the unison D in m. 19, written in the score as A, D# and F# in the viola, violin and flute. Most of the salient musical surface in the first 20 measures is governed by the 3cycle, and this trichord in m. 19 is the first non-unison simultaneity of the work. Once the D is introduced in m. 20, the following sonorities belong to the 5-cycle: notice the chord in m. 22 with the D and A in cello and violin, the D# in the viola first paired with the Ab in the flute, then with the Bb that immediately follows.

Figure 9-12, the non-unison simultaneities in mm. 12-22. An incomplete ic3 cycle emerges within the ic5 cycle

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Let us now consider the unison simultaneities, for they outline a deeper unfolding cycle that governs the large-scale form of the first section of the piece. There have been 3 full-ensemble unisons up to this moment: the Ab in m. 12, the D in m. 20 and the B in m. 32. The 3-cycle is revealed, and pc5 is expected for its full completion. If we dare to predict its appearance following the proportions offered until this moment (a unison simultaneity every 10 measures, or so) we can expect pc5 at around m. 42. Yet Davidovsky has other plans for m. 42. Figure 9-13a shows the moment in which the chromatic (01234) chord occurs on the downbeat of m. 42. Figure 9-13b maps it within the 1-cycle. Pitch C is singled out in the flute with a slight 16th note delay, and with a sforzando-piano dynamic.

Figure 9-13a, the chromatic pentachord in m. 42, where a unison pc5 could have been expected to complete the unfolding 3-cycle within the full-ensemble unisons

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Figure 9-13b, the (01234) chromatic pentachord mapped within the 1-cycle

Figure 9-13c maps the unfolding 5-cycle gesture that opened the piece and its chromatic transformation via the M5 relationship. The chromatic pentachord is not only the first forte dynamic of the work; it is also the first (of many) chromatic chords to follow. Yet also shown in figure 9-13c is the chord’s missing pc5 that would fully transform the opening 5-cycle gesture from C to Db into a (012345) chromatic hexachord. Not only is pc5 missing from the chord, it is also the pitch that we were expecting in order to complete the unfolding 3-cycle. So, the question is - where is pc5?

Figure 9-13c, initial 5-cycle gesture from pc0 to pc1 transformed via M5 into the (01234) chromatic pentachord at m. 42. Pc1 (which had a weighted presence as both Db and C# were present in the music) is missing its partner, pc5

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Figures 9-14a and 9-14b show the build towards the violent, fullensemble unison F that finally occurs on the downbeat of m. 68, which marks the end of the first large section of the work. There are four strong attacks starting on m. 63, marked double sforzando. All four are incomplete unisons, as the violin is busy playing a rapid sextuplet figure: E on m. 63, C and Db on m. 64, and D# in m. 65. These pitches can be isolated and paired as two ic3 dyads a half step apart, and they establish symmetry around D. We will call this symmetry SymI4. Mapping these incomplete unisons to our already unfolding 3-cycle of full-ensemble unisons within our space, the F’s absence seems even more evident. Indeed, this might be Davidovsky’s notion of a cadence. In m. 66, a new chromatic chord fills the 1-cycle space, this time, a (0123) tetrachord. The importance of this chord is twofold. Firstly, this is the M5 chromatic transformation of the 5-cycle gesture heard from mm. 20-36, which unfolded on the left side of the 5-cycle towards the faint C# in m. 36. Secondly, this chord, as the one previously heard in m. 42 (which was, in turn, the chromatic transformation of the opening 5-cycle gesture) is also missing pc5 for it to be a complete transformation. Finally, in m. 68, Davidovsky provides us the long anticipated F, in a violent con tutta forza unison, so clear and crisp that it cannot be missed. This F fulfills several motions: first and foremost it completes the slowly unfolding 3-cycle created by the full ensemble unisons that started in m. 12; secondly, it fulfills the complete M5 transformations of both opening ic5 gestures (the original, from C to Db in mm. 1-15 and its mirrored response from D to C# in mm. 20-36). As we saw in Figure 9-13b, the chromatic transformation of the opening gesture occurred in m. 42, and it was missing pitch F, as was the chromatic chord in m. 66. Thirdly, it fulfills SymI4, around pc2, which had been governing the deeper motion towards the unison F since m. 12. This symmetry is placed in opposition to the initial SymI1 and Sym3, which governed the more salient musical motions of the first section of the piece.

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Figure 9-14a, leading up to the unison pc5 in m. 68

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Figure 9-14b, two main events are mapped within the outer 5-cycle: the (0134) tetrachord (the incomplete unisons heard in mm. 63-65), and the Ab-B-D fullensemble unisons that have been unfolding the 3-cycle since m. 12.

Figure 9-14c, one new event is mapped within the inner 1-cycle and added to our space: the chromatic (0123) tetrachord in m. 66, the M5 transformation of our mirrored ic5 gesture from mm. 20-36; pc1 is missing its chromatic companion, pc5

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I hope to have shed some light on Davidovsky’s approach to pitch in this chapter, and how his musical motions can be interpreted through interval cycles, symmetrical collections and aggregate completions. As I mentioned in the introduction, his pitch organization is based on a cyclical conception of the aggregate. Different layers of cycles help shape the deeper formal structures and salient features of the music, including his rhetorical phraseology and cadential motions, to name just two of the most-written about aspects of his music.4 That symmetrical collections are an important element of Davidovsky’s aggregate completion technique is made clear in the several examples presented here. Yet some of the issues I brought up need further investigation: the relationship (being that of concordance or discordance) between axes of symmetry and salient pitches, and the idea of symmetry expectation being the most notorious ones. .

Notes 1

I would like to provide a warm thank you to my advisor, Joseph Straus, for steadily guiding me through my research, and to Jeff Nichols for all his support and advice along the way. 2 There might be a physical reason why this is so: all four Quartettos were written for a solo instrument and string trio. We might speculate that the physical construction of the string instruments (tuned in fourths and fifths) could have influenced Davidovsky’s pre-compositional process. He is also a talented violin player and ic5 could, indeed, be also deeply embedded in his mechanics as both performer and composer. 3 See, for example, Hubert Howe’s "Some Combinational Properties of Pitch Structures," Perspectives of New Music, Vol. 4, No. 1 (1965): 54-61. 4 Two dissertations are to be mentioned here: “Angled Responses: Structural Pairing in Mario Davidovsky's Quartetto” by Steven Leon Ricks (University of Utah) and “Dialogue, Narrative, and Other Behaviors in Mario Davidovsky’s Quartetto” by Justin A. Rust (Brandeis University).

CHAPTER TEN FORMAL, IMPULSE, AND NETWORK STRUCTURES IN DONALD MARTINO’S IMPROMPTU NO. 6 AARON J. KIRSCHNER

In his 1991 analysis of Donald Martino’s Impromptu No. 6, Henry Klumpenhouwer focused primarily on local trichordal relationships within the first section. While his analysis of these details was quite thorough, and served to demonstrate the power of K-Nets, it did not address larger formal structures articulated by the pitch material. This chapter will focus primarily on those larger structures, as well as the use of meter and rhythm to delineate the form. In particular, my analysis will examine the voicing of trichords as a formal procedure, the development of rhythmic density, and the relationships between trichords within an aggregate and the relationship of aggregates in a group of four. The sixth impromptu is constructed primarily from (014) trichords. Save for the hexachords in bars 9–14, each discrete set of three pitches is a member of set-class (014). Furthermore, each statement of four trichords presents a complete aggregate. There is neither infiltration nor retrogression within statements of a row-form—any pitch-class presented will not be stated again (excepting immediate repetition) until its associated aggregate has been presented in full. The grouping of these trichords into aggregates is shown on the score in Example 10-1. Each bracketed letter corresponds to a different aggregate, of which there are 13 distinct forms. Throughout this paper, the aggregates are referred to by their letter, and the trichords are referred to by their aggregate letter and an Arabic numeral referring to the trichord’s ordinal position (i.e. trichord D3). The final two aggregates are repetitions of aggregate M, hence the use of superscript Arabic numerals. All aggregates except letter I can be quartered into subsets of ordered trichords (letter I splitting into complementary hexachords). Removing letter I, and

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not counting the repetition at M2 and M3, shows a total of 48 ordered trichords. Full Score Aggregates labled by letter

A

B

C

E

G

F

(I)

H

J

L

D

M1

K

M2

M3

Example 10-1, full score of Martino, Impromptu No. 6, with aggregates labeled by letter

Beyond the simple fact that each discrete trichord is of set-class (014), the vast majority (43 of 48, or about 90%) are presented in a voice

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structure that will be referred to as “i11-enclosed.” This describes the boundary interval—the third pitch is enclosed within two pitches a major seventh apart.1 The other possibilities for any (014) trichord (when reduced to the smallest intervals that still maintain chord voicing) are i4enclosed, i13-enclosed, and i15-enclosed. Of the five trichords that do not exhibit i11-enclosure, three (E1, F4, L1) exhibit i15-enclosure and two (E3, L3) exhibit i4-enclosure. The lack of i13-enclosure is not insignificant—it emphasizes the importance of i11-enclosure insofar as when that is not exhibited, it is not replaced with its complement. Given the abundance of i11-enclosure, it shall be considered the “normal” structure of the trichord—for simplicity, “unenclosed” will refer to any trichord that does not exhibit i11-enclosure. In addition to the voicing of the chords, it is important to note the difference between chords presented in a strict block-chord fashion and those that are more “lyrical.” (For the purposes of this analysis, “lyrical” will refer to any trichord that is not stated with all three pitches simultaneously.) The majority of trichords (33 of 48, or 68.75%) are presented in block chord fashion, although the metric spacing of the lyrical chords makes this less obvious from a durational perspective. Interestingly, of the five trichords that do not exhibit i11-enclosure, four are also presented lyrically (E3, F4, L1, L3). The remaining trichord (E1) is not itself lyrical, but is given such an effect, as it is approached by grace-notes of the previous trichord, which are metrically offset from the previous chord and registrally related to E1.2 Thus, it is clear via percentages that i11-enclosed block chords are the primary element of the Impromptu. That the work opens and closes with these elements further reinforces their importance. Figure 10-1 presents a table of unenclosed and/or lyrical trichords.

Unenclosed

Lyrical

E1, E3, F4, L1, L3

D4, E2, E3, F1–4, G1, G2, K2, K4, L1–4

Unenclosed & Lyrical E3, F4, L1–3

Figure 10-1, table of trichords that are unenclosed and/or lyrical

The Impromptu formally divides into three main sections, which coincide with statements of the aggregate. Measures 0–4 comprise the first section, and consist entirely of registrally-disjunct block trichords of i11enclosure. Measures 4–9 comprise the second section, with 10–14 acting

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as a transition into the last section. These measures are the most lyrical— nine of the fifteen lyrical trichords occur in bars 4–9. The transition presents the only material not explicitly derived from (014) trichords, instead presenting rapidly alternating complementary hexachords. The third and final section, from measure 15 to the end, closes the piece.3 It begins and ends with block-chords of i11-enclosure, with lyrical statements in the middle—exactly how the whole piece is structured. Having established this formal design, it is important to observe the significance of those trichords that are lyrical and/or unenclosed. A glance back at Figure 10-1 shows that these chords lie predominantly in the second section of the work, with the rest in aggregates K and L (which relate formally back to the second section). Thus, in the most general terms, the piece tracks the development of block chords to lyrical lines and back again. The brief return to lyrical chords at L does not frustrate this development. Rather, it reinforces the form by creating a smaller version of the large-scale structure and brings back material from the middle section in the closing material—one of the many references to 19th century piano literature found throughout the Fantasies and Impromptus.4 Furthermore, there is an interesting dichotomy when the ending trichord of a section is compared with the structure that is generally exhibited by all trichords in that section. The first section comprises aggregates A–D, all of which are i11-enclosed chords. Furthermore, all but D4 are block-chords—it is the single grace note at the beginning of bar 4 which separates D4 from all other chords in its section. A similar concept is seen in aggregate H. Aggregates E–G are the main material of the much more lyrical second section, and while H is clearly part of this section, it is entirely block i11-enclosed trichords. Thus, while the two first sections can clearly be delineated as block-chords versus lyrical passages, their endings exhibit the reverse. This cannot be a coincidence, but rather a means of connecting the sections5 (section 3 begins with block-chords, after a block-chord transition at aggregate I). *** In addition to set-class voice structure, metrical elements play a defining role in the form. Figure 10-2 presents an Impulse-Register graph of the Impromptu. The note-heads represent attacks and their position on the staff lines represent the register—low, middle, and high.6 The aggregates are labeled by letter beneath the first attack of each aggregate; slurs connect impulses of the same trichord.

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M3

Figure 10-2, Impulse-Register graph of Impromptu No. 6

The opening section establishes the eighth-note as the primary pulse, and to this all other pulse units can be related.7 For the moment, the pulse relationships of the first section will be considered. Having established the eighth-note pulse, there is a clear relationship between aggregates A and B. Both begin with three eighth notes, which are followed by a longer value; the effect is a three eighth-note pick-up into a downbeat. Aggregate C reverses this structure, with the longer note value preceding the three eighths. By contrast to all of these, aggregate D splits the eighth-note impulses between two dotted eighths. The subtle shifts from simple to compound meter serve to obfuscate the higher beat units, and thus reinforce the eighth-note as pulse. Moreover, these metrical relationships allow a certain written-out pushing and slackening of the tempo, while keeping the eighth note constant. Observe the relationship between the half note in bar 1 and the two dotted eighths in bar 2. Here, while the motion is clearly slower than the eighthnote pulses around it, the two dotted quarters have a sense of pushing the tempo due to their relationship with the half note that precedes them. While some analyses have considered this a form of syncopation,8 whereby the performer should treat the second dotted quarter as an upbeat, the notation of the passage suggests otherwise. Had Martino intended this to be treated as an upbeat, he most likely would have notated the measure in #4as q-ry-q rather than in the notated ^8. It is almost certain—given Martino’s specificity in notation9—he would have at the very least included a ˇ sign to make sure it was treated as an upbeat. While the relationship between bars 1 and 2 is that of pushing forward, the relationship in bar three is one of slackening. In this case, the eighthnote pulse becomes dotted eighths. While this technically could be

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considered syncopation by the above standard, the use of dotted eighths in a compound meter to signify a duple subdivision is much more common (and more easily understood by the performer) than using ^8to imply syncopation. And Martino’s notation at the end of this Impromptu clearly demonstrates that he considers these two dotted eighths following four eighths to be the same as the hyper-specific “ @4 plus ^6” marking that is implied. Beyond such local relationships, the impulse structure also creates larger formal implications. These are directly tied to the movement from block-chords to lyrical passages. By simple math, it requires more impulses to state an aggregate as individual pitches than as ordered trichords. Thus, as the music changes from a block-chord texture to a more lyrical one, there will—by definition—be more impulses per aggregate. If the rate of impulse is kept the same, then a pitch-by-pitch statement will take three times as long as an ordered trichord. Conversely, to keep the duration of the aggregate presentation the same, the pitch-by-pitch statement would require an impulse density of three times that of the ordered trichord statement. Figure 10-3 demonstrates these relationships with hypothetical derivations of aggregate A.

Figure 10-3, two hypothetical derivations of aggregate A into lyrical structures

As shown above, triplet sixteenth notes are required should a completely lyrical statement of the aggregate be accomplished in the same duration as a trichordal statement in eighth notes. Such a note value is indeed reached at the end of bar 6, but this is well after the music has become lyrical. Moreover, it is not an immediate shift to the triplet sixteenths—they are preceded by regular (duple) sixteenths. The result is that the aggregate is being presented at a slower rate, even though the rhythms themselves have sped up. Equally interesting is the placement of a

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dyad on the second triplet sixteenth of bar 6. This is the first impulse of aggregate G, and the third triplet sixteenth-note completes the trichord G1. Significantly, this means that trichord G1, while part of a rhythmic unit that would allow a lyrical statement of the aggregate in the same time established in aggregate A (and B), completes itself even faster than the rhythmic diminution would imply. While the trichordal statements in aggregate F presented the aggregate at a slower rate than the opening, G1 is faster. Whole number ratios10 in the rates of aggregate presentation and/or impulse density with relation to the opening are eschewed in favor of more complex relationships. Impulse relationships also relate the third section and the opening. Aggregate J (bar 15) exhibits a very similar metrical structure to the opening, with three eighth-note impulses followed by a longer note value. In this case, the longer value is a quarter. This is significant in two ways. First, it continues (and concludes) the diminution of the final note value that was established between aggregates A and B—what was a half becomes a dotted quarter becomes a quarter. Should this diminution continue, the longer pulse would become a simple eighth-note and be indistinguishable (rhythmically speaking) from the three preceding values. The second observation to be considered is the relationship between bars 3 and 15, the only &8measures in the short piece. Both share the common thread of beginning with steady eighth notes, followed by longer note values—the difference lies in the longer values themselves. While in the first &8bar the slackening of the pulse was by augmentation of a sixteenth note to the longer values, here it is by addition of a whole eighthnote. What were dotted eighths become quarters. To make this fit in a &8 measure, an eighth must be left off of the opening pulse. Aggregate J, as the first aggregate of the final section, is marked in its relationship to the opening section by both augmentation and diminution. Doubly notable, it prevents a true whole number ratio in terms of aggregate J’s relationship to the opening material—it comes very close, but does not quite reach that level of similarity. Further in this section, note the steady dotted eighth notes in the three repetitions of aggregate M (mm. 18–20). In the coming section, the strong relationships between aggregate M and the opening will be made clear from a pitch-class standpoint, but their metrical similarities are equally meaningful. The dotted quarter pulse throughout these is clearly an augmentation of the opening pulse, creating a ratio of 3:2. What is missing is the arrival and resting on the fourth impulse—a straight augmentation would give e. e. e. h.. But, as seen in aggregate J, the diminution of that final beat has already been completed. Continuing the diminution would simply make the longer value the same as the shorter ones—exactly what

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is being done with this pulse, save for the augmentation to the whole of it. Thus, this third and final section combines the elements of augmentation and diminution that operated independently in the opening section. While the metrical analysis up to this point has explored relationships of fractional ratios, there is one section that exhibits whole number ratios to the opening impulse structure. Made painfully obvious by the sheer amount of black ink needed to engrave it, aggregate I (mm. 9–14) is clearly a 2:1 impulse rate from the opening. Moreover, because of its alternating complementary hexachords, the entire aggregate is stated every two impulses, or every eighth note. Thus, while the rate of impulse is double that of the opening, the rate of aggregate presentation is quadrupled. In the one section where the relationship of ordered (014) trichords is less structural, the impulse structure is given its clearest relationship. Moreover, these whole number relationships are markedly different than those exhibited in the other sections—almost all of the notable examples included a relationship by 3, while these are relationships by 2. *** Having established the importance of the aggregate structures, attention will now be turned to the relationships between formally significant trichords. There is not a clear row form in this piece, at least not one that can be teased out as easily as it could in the first of the Fantasies.11 There are, however, very strong relationships between each trichord, and not simply because they are all of the same set-class. This analysis will make use of Klumpenhouwer networks to explain the relationships and voice leading between notable trichords. Appendix 1 contains the K-nets for each aggregate and a hyper-K-Net for sections one and three (aggregates A–D and J–M). Aggregate A comprises the majority of Klumpenhouwer’s 1991 analysis. He demonstrates that registrally adjacent trichords are strongly isographic while those metrically adjacent—but registrally separated—are related by the hyper-transformation (i.e. isographic by the automorphism) 12 < T 8> . I will use this aggregate to review some elements of K-net transformation and explain my methodology. First, I have privileged the T8 transposition vector in each (014) trichordal K-net. This stems from the importance of the i11-enclosure of the trichords, and that T8 represents the widest space between any two pitches in an i11-enclosed trichord, and thus its position (either at the top or bottom) is quite aurally apparent. Because the vast majority of the trichords fit this model, they will thus be related by some form of positive isography. This means that there is some hypertransformation that will map the pitch-transformations from one i11-

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enclosed (014) trichord onto another. Those trichords whose transformations map onto each other without a hyper-operation (stated as ) are called strongly isographic. A byproduct of this property is that each pitch-class from one trichord can be transposed by a specific interval (but not specific direction) onto a discrete pitch in another trichord it is strongly isographic to. These relationships can be observed in Figures 104a and 10-4b showing aggregate A. Note that the hyper-T-operand means that the pitch-level transposition is held constant, while the axes of inversion are rotated by the value of the hyper-operand. Also demonstrated, in Figure 10-4b, are the perfect-fifth transpositions between each pitch-class of strongly isographic trichords.

Figures 10-4a-b, aggregate A shown as a K-net, and its voice leading by perfect fifth

The remaining aggregates in the first section exhibit similar isographic relationships, with registrally-similar chords being strongly isographic and

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metrically adjacent chords exhibiting positive isography. Klumpenhouwer provides a more than satisfactory analysis of each trichord in this passage,13 but it is also worthwhile to consider the relationships between each aggregate. Notably, within any given aggregate from A–D, each trichord is related to any other by one of the three following automorphisms: or . Moreover, each aggregate contains two pairs of strongly isographic trichords, and of the remaining four automorphisms, three are and the other or vice versa. Figure 10-5 summarizes these relationships in a hyper-K-net, showing that each aggregate is strongly isographic (either positively or negatively) to each other via hyper-automorphisms. While the automorphisms do not directly propagate upward (as there are no or hyperautomorphisms), the arrangement of the aggregates parallels the trichords insofar as there are two pairs of strongly positively isographic networks, with the remaining related by complementary operations (considering that is self-complementary).

Figure 10-5, Hyper-K-Net showing aggregates A–D with hyper-hypertransformations

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Before exploring the second section, attention will be turned to aggregate A’s formal mirror, aggregate M, as its relationships are directly related to those of the first section. Referring to these as mirror relationships does not simply refer to their formal position, but also to the low register chords. These are exact retrogrades of aggregate A—A1=M3, A3=M1 in both pitch and pitch-class space. The upper register trichords are different, and their spatial relationship is “prime” rather than retrograde. This change in the upper register trichords means that the network relationships within the aggregate are also changed. In fact, in this aggregate, all trichords are strongly isographic. Figure 10-6 demonstrates these relationships.

Figure 10-6, K-Net of Aggregate M

Moreover, not only this aggregate, but all aggregates of the final section (J–M) exhibit this “all-isographic” property. Aggregate J is an outlier in terms of its specific voicing—its unenclosed structures create certain negative isographies via the automorphism when the T4 transformation is considered. However, is the retrograde automorphism of , which is itself relevant insofar as it converts the T4 transformation to the T8 transformation that has proved a vital component of the harmony. Figure 10-7 provides the hyper network for this section— notice how the “all-isographic” arrangement propagates upward. While the opening section exhibited similarities in the automorphisms and hyper-

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automorphisms, the final section presents an even stronger link between the two.

Figure 10-7, Hyper-K-Net transformations

showing

aggregates

J–M

with

hyper-hyper-

Figure 10-8, voice leading of Aggregate M

The “all-isographic” collection of trichords at aggregate M provides a plethora of voice-leading implications. First, the relationship between M1– M3 is identical to that from A1–A3, with the perfect fifth transposition vectors’ direction reversed due to the retrograde. M2–M4 also exhibit perfect fifth transposition vectors. Two of the A1–A3 vectors are of negative direction, while A2–A4 presents two positive directed vectors.

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M1–M2 and M3–M4 share the same set of transposition vectors, all by semitone, with two negative and one positive. Finally, M1–M4 and M2– M3 are voice-led via tritone transposition vectors. These vectors are not given a direction because in mod12 space a vector of magnitude 6 is indistinguishable in its positive and negative forms.14 Figure 10-8 demonstrates the voice leading for each of these vector relationships. In each case, the first chord listed is placed in its registral arrangement and the second is placed in its isographic configuration with the first chord. They demonstrate that any chord in aggregate M can be related to any other by transposition of each pitch-class by a vector of specific magnitude. The formal significance of this strong isography should not be underestimated. While the first aggregates were presented with strong isography between registrally adjacent trichords, the ending section connects every chord. This also links to the impulse structure discussed in the previous section—the final aggregate is the only point where all impulses are of the same unit, and it is concludes the section where all trichords and aggregates are strongly isographic. In terms of both pulse and automorphism, no place in this aggregate is distinguishable from any other. While there are many other network associations that can be drawn (especially the registral changes in aggregate H), my analysis will now turn to the material in aggregate I (measures 9–14). Here trichords are abandoned in favor of complementary hexachords. These hexachords are not made up of two (014) trichords, so it would seem that this is simply different material rather than some extension of the previous networks. However, the boundary pitches of these hexachords are worth noting. A minor sixth binds each hexachord—the lower from G–Eb,15 the upper from Bb–Gb. These pitches do form two (014) trichords, both with G and Gb as the (01) subset. These trichords are inversionally related by the index I1 and isographic via the automorphism (see Figure 10-9). This is the only negative isographic relationship between i11-enclosed trichords in the piece, one of the elements making these measures so distinct from the rest of the music. However, just as with the whole number ratios in the impulse structure, there are clear connections even in this seemingly anomalous section.

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Figure 10-9, extracted trichords of Aggregate I

Were this the only relationship between trichords sharing two pitchclasses, it might be of little relation to the rest of the structure. However, a return to the opening section dispels that doubt. Observe the opening trichords of aggregates A–D. Each shares two pitch-classes with the chord preceding it, configured such that what was the bottom-most pitch becomes the middle pitch. From A1–B1 and C1–D1, the common pitches are the (04) subset of the chord. This fits perfectly into the second general formula for (014) trichords sharing two pitch-classes. Thus, each of these trichords is isographic with the following via the automorphism . Figure 10-10 demonstrates the relationships between the first trichords of aggregates A–D.

Figure 10-10, K-Net of A1, B1, C1, D1

Each of the hyper-transformations in these relationships have formal significance. The automorphism is common to the Impromptu, yet its significance here lies in its existence only between the first and last trichords. Given that the arrival of trichords of strong isography signals the end of the work, this is a subtle method of creating a composed-out form.

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The first and last aggregates of the opening section have a similar relationship to the structure of the entire piece (much as the lyrical structure of aggregate J reinforces the form). The and hypertransformations are more specific to this configuration, and allow two possible understandings of the voice leading from A1–B1. The first sees the two upper pitch-classes shift upwards by semitone, while the third moves up by eight semitones, changing its voicing from the bottom of the chord to the top. This demonstrates a shift in registral spacing, but similar motion in the voice leading. The other, equally possible, voice leading involves the bottom two pitch-classes moving by octave, while the uppermost shifts down by 10 semitones, creating the same shift in voicing, but only transposing one pitch. Both of these relationships are demonstrated in figure 10-11.

Figure 10-11, two possible understandings of the voice leading from A1-B1

*** The importance of each of these examples is that they show different degrees of relationship between trichords of the same set-class. To simply analyze each of the trichords as (014) and assign an operand between it and the next would be to provide a “correct” analysis while ascribing nothing to the musical performance aspects of the work. Likewise, such operands do little to link the music to the well-defined impulse and formal structure that is so important to the cohesion of this short piece. These networks not only work within the formal structures of the piece, they reinforce them. The journey from block-chord to lyrical and back is accompanied by a journey through different degrees of isography, ending with the strongest isography of all. Likewise, this isographic journey is accompanied by a journey through augmentation and diminution of impulse. These elements are not disparate, but inexorably intertwined.

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Appendix 1: K-Nets for each aggregate

A

B

C

D

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F

G

H

I

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J

K

L

M

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D

J
M

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Notes 1

Vishio, 1991, 454–6. Dembski, 1991, 315. 3 Klumpenhouwer, 1991, 318–20. 4 Martino, 1991, §1. 5 Stadelman, 1991, 419–20. 6 Klumpenhouwer, 1991, 323–5 (This article contains a very through analysis of the role of register). 7 Ferneyhough, 1989, 54–5. 8 Klumpenhouwer, 1991, 324. 9 Martino, 1966, 50–2. 10 Ratios (or their multiplicative inverse) that divide evenly into whole numbers (i.e. 3:1 or 2:6, but not 4:3). 11 Vishio, 1991, 443–4. 12 Klumpenhouwer, 1991, 329–33. 13 Ibid., 335–42. 14 Vishio, 1991, 359–64. 15 The hand crossing and clef switch is simply for ease of performance. 2

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Works Cited Dembski, Stephen. “Misreading Martino.” Perspectives of New Music, Vol. 29, No. 2 (Summer, 1991): 312–317. Ferneyhough, Brian. “Duration and Rhythm as Compositional Resources.” In Brian Ferneyhough—Collected Writings, ed. James Boros and Richard Toop. Amsterdam: Harwood Academic Publishers GmbH, 1995, 51–65. First presented as a lecture by the author in 1989, at the National Percussion Conference in Nashville. Klumpenhouwer, Henry. “Aspects of Row Structure and Harmony in Martino’s Impromptu Number 6.” Perspectives of New Music, Vol. 29, No. 2 (Summer, 1991): 318–354. Lewin, David. “Klumpenhouwer Networks and Some Isographies that Involve Them.” Music Theory Spectrum, Vol. 12, No. 1 (Spring, 1990): 83–120. Martino, Donald. A Conversation with Bruce Duffie. Telephone interview conducted by Bruce Duffie. WNIB, January 12, 1991. Transcript, 2009. Accessed April 1, 2013, http://www.bruceduffie.com/martino.html —. “Notation in General—Articulation in Particular.” Perspectives of New Music, Vol. 4, No. 2 (Spring–Summer, 1996): 47–58. Stadelman, Jeffery. “A Symmetry of Thought.” Perspectives of New Music, Vol. 29, No. 2 (Summer, 1991): 402–439. Vishio, Anton. “An Investigation of Structure and Experience in Martino Space.” Perspectives of New Music, Vol. 29, No. 2 (Summer, 1991): 440–476.

CHAPTER ELEVEN EXPLORING NEW PATHS THROUGH THE M ATRIX IN URSULA M AMLOK’S FIVE INTERMEZZI FOR GUITAR SOLO ADAM SHANLEY

In Ursula Mamlok’s Five Intermezzi for guitar solo, written between 1984 and 1990, the composer creates a suite of five separate pieces that combine to make a unified work exploring diverse textures, intricate harmonic relations and varied melodic lines from a single 12-tone matrix. Each intermezzo stands as its own structurally sound unit, and as an integral part of the work’s overall arch form. In this chapter I explore in detail some of the many strategies by which Mamlok creates unity across the entire work as well as within each individual intermezzo. My approach takes into account not only pitch relations, but also formal and rhythmic elements. First I will discuss properties of the matrix, then I will discuss elements of each intermezzo in turn, uncovering their harmonic, melodic, rhythmic, formal, and motivic inner workings. In the course of these individual analyses the form of the work as a five-movement entity will also be revealed. The many layers of the composition reveal a structure that is dependent upon unique characteristics of the matrix that are brought out by Mamlok via a spiraling path that guides the work to the center and back out again. Surface level details are found to be deeply rooted in the construction of the initial row form. Scholarly study of Ursula Mamlok’s works has only begun recently, with Roxane Lise Prevost’s 2003 doctoral dissertation leading the charge.1 Prevost’s research outlines the compositional and stylistic elements of Mamlok’s serial works that began with her Variations for Solo Flute of 1961 and continue through to her more recent works. Many of the compositional strategies uncovered by Prevost also appear in the Five Intermezzi, such as the use of recurring motives, cyclic construction, tonal references and a non-dogmatic approach to pitch generation. Prevost’s

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extensive interviews with the composer also provide us with many helpful insights into her compositional process. Ursula Mamlok was born in 1923, emigrated to the United States in the late 1930’s and studied with Stefan Wolpe, Ralph Shapey and Roger Sessions in the early 1960s. According to Joseph Straus: Mamlok’s own twelve-tone language has often involved taking unusual musical paths through the familiar twelve-by-twelve matrix of row forms. In some works, for example, the musical lines trace a spiral path through the matrix, a succession of diminishing concentric squares culminating in 2 the notes that lie at the center of the matrix.

These “unusual musical paths” through the matrix, including the “spiraling concentric squares” and the ideas of motivic repetition and cross-reference that Straus mentions, are some of the techniques at work throughout these intermezzi that allow Mamlok to break free from the matrix altogether when appropriate. Since the work is comprised of five pieces, it allows for an arch form schematic to be the groundwork upon which the piece is built. The third intermezzo acts not only as the actual centerpiece of the work with two intermezzi on either side, but the center of that central movement becomes in many ways the focal point of the work.

Properties of the Five Intermezzi Matrix The first step is to examine the construction of the matrix. The first nine notes of Mamlok’s row consist of three adjacent [036] 3-10 trichords. Of course, in tonal language, we would equate these to diminished triads, with the remaining pitches creating an [015] 3-4 trichord.

Figure 11-1, Ursula Mamlok Five Intermezzi row

The row’s construction allows for several patterns at various levels of the matrix to come forward. Dividing each hexachordal quadrant of the matrix into quadrants of trichords, shown by the dashed lines in Figure 11-2 below, reveals to us that all trichordal quadrants except for the trichords belonging to rows P5 through P0, and those belonging to rows I1 through I6, encapsulate collections of [0369] 4-28, or fully diminished tetrachords. The aforementioned rows whose quadrants do not encapsulate pcs from a

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single 4-28 tetrachord, in fact contain trichord subsets from each of the three possible tetrachords, organized in a specific pattern, described below.

Figure 11-2, Five Intermezzi matrix. Dashed lines divide each solid-line hexachordal quadrant into trichordal quadrants.

The odd quadrant out is that which falls in the bottom right-hand corner, the last trichord of rows R5 through R0. This quadrant forms [015] 3-4 trichords for each vertical and horizontal consideration. For example {9, 1, 2}, {5, 9, t}, and {4, 8, 9}, found horizontally, as well as {9, 5, 4}, {1, 9, 8}, and {2, t, 9} vertically are each members of set class 3-4. Closer inspection in fact uncovers this seemingly odd quadrant as containing a much-condensed version of the rest of the matrix. If we take a step back from the single pitch level of the matrix and begin to look at patterns created by the pitch collections, yet another layer of organization is revealed. Since we know that the first three trichordal quadrants from left to right and the first three trichordal quadrants from top to bottom contain 4-28 trichords, we can examine the pattern created by their organization in Figure 11-3 below. Shown in the figure is an analysis of the 4-28 tetrachordal collections and their subsets contained in the matrix. Numbers in brackets refer to the particular 4-28 collection that is contained within the quadrant. For example {0} refers to {0, 3, 6, 9}, {1} to {1, 4, 7, t} and {2} to {2, 5, 8, e}. The far right quadrants and the bottommost quadrants include 3-10 subsets from each 4-28 tetrachord in rotating patterns.

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Figure 11-3, Pattern of 4-28s and 4-28 subsets within the Five Intermezzi matrix

Looking horizontally across the bottom trichordal quadrants, for example, we see that the trichordal segments of P5 are taken from 4-28 tetrachords built from pcs 2, 1, and 0, respectively, while individual pitches found in the final quadrant are taken from 4-28 tetrachords built from pcs 0, 1, and 2, respectively. This pattern creates the palindrome 2-10-0-1-2, and looking at the remaining rows, P1 and P0, we can see that they similarly form their own palindromes. This is also true for the trichordal segment of rows It, I5, and I6. In actuality the patterns exist across all quadrants, as is shown in Figure 11-3 above. Finally, regarding these patterns derived from 4-28 tetrachords and their subsets, rotations taken from the palindromic patterns uncover a larger palindrome derived from the patterns of subsets. For example if we look down the far right quadrant, but reading each trichord from left to right we can see this pattern: 1-2-0/2-0-1/0-1-2/0-1-2/2-0-1/1-2-0. Of course, a similar pattern is created through the rotation that exists across the trichord segments of the bottom three prime forms. At a higher level of structure, the matrix also displays the property of hexachordal retrograde-inversional combinatoriality, hereafter HRIC, as can be seen in Figure 11-4 below. Note also that the pitches in the first hexachord of RI2 are a T4 order-number transposition of those in the second hexachord of P9. The HRIC property allows for what I am calling a “spiraling” characteristic of hexachordal invariance. I am using the term to describe the movement of Px hexachords down their hexachordal quadrant while their matching invariant hexachord moves from right to left through the same hexachordal quadrant.

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Figure 11-4, Hexachordal retrograde inversional combinatoriality within the Five Intermezzi matrix

If we look back to Figure 11-2 we can see that the first hexachords of P9 and I2 have the same pitch content, and because of the property mentioned in the previous paragraph, the first hexachord of P9 is identical in content and order to the second hexachord of RI2. This same property also exists between the first hexachords of P6, P3, Pt, P7, and P4 and the second hexachords of RIe, RI8, RI3, RI0, and RI9, respectively. This “spiraling” characteristic is only true of the NW and SE hexachordal quadrants of the matrix. These properties of spiraling hexachordal invariance and HRIC are what allow Mamlok to trace a unique path through the matrix while allowing for recurring harmonic and melodic content on the surface of each intermezzo. I will now show the way these specific characteristics that make up the matrix form the basis for each intermezzo, as well as the structure of the five intermezzi in its entirety.

Intermezzo I: Grazioso

Copyright © 1992 by C.F. Peters Corporation. Used by permission. Example 11-1, Intermezzo I, mm. 1-3, showing order numbers for P9, and It. Order numbers for the rows are shown above the staff.

Example 11-1 above shows the opening of the first intermezzo, which contains a simple linear phrase that arcs upwards through P9 then descends through It.

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Copyright © 1992 by C.F. Peters Corporation. Used by permission. Example 11-2, Intermezzo I, mm. 4-8, showing the continuation of It followed by RIt. Order numbers for the rows are shown above the staff, and to the left of any simultaneities.

Measures 4-8 introduce a contrasting legato phrase that exposes the second hexachord of row-form It. This phrase features a repeated descending pc6 to pc2 gesture in the upper voice, transitioning to RIt in measure five via a pc1 grace note pivot. The 4-18 tetrachord that closes the phrase, and section, is marked by a sforzando thud on the body of the guitar similar to that which opens the piece. Measures 4-6 in this passage add to the repeated melodic content a set of two overlapping rhythmic palindromes divided between the upper pc6 and lower pc2. Discounting the grace notes. the voices break down rhythmically as shown in Figure 11-5 below. The legato nature of this contrasting phrase is exaggerated through the disorienting denial of a regular pulse. Its pattern of eighth, dotted-eighth, and sixteenth notes, all of which are tied over the bar-line, belies the 2/4 meter entirely. Figure 115 simplifies the rhythmic structure by grouping attacks together with the sixteenth note pulse as the common denominator, and by forgoing the barlines. An underlying symmetrical rhythmic organization with the eighth-note pc2 being placed in the exact middle of the pattern is thus revealed.

Figure 11-5, Intermezzo 1, mm. 4-6, rhythm simplified

Measures 16-22 closely resemble the design of measures 4-7, with a repeated gesture in the upper voice that elongates the row’s unfolding. The repeated descending major 3rd of measures 4-8 is not only mirrored in these later measures of the first intermezzo, but we will see that this figure

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returns in later intermezzi, true to Roxane Lise Prevost’s findings regarding Mamlok’s use of motivic repetition and cross-reference within her works. This is shown in Example 11-3 below.

Copyright © 1992 by C.F. Peters Corporation. Used by permission. Example 11-3, Intermezzo 1, mm. 16-22 drawing piece to a close with row-form Ie. Order numbers are shown above the staff, and to the left of any simultaneities.

Figure 11-6, Intermezzo 1, mm. 16-22 with simplified rhythm. Numbers below the staff are the number of 16th notes in each tremolo pitch showing the gradual diminution of rhythmic values.

This closing passage is similar to measures 4-7 in its rhythmic design as well. As a way of creating contrast between sections, it is also noteworthy that the changing meters within this intermezzo find measures of compound meter being rigid in their containing only 16th notes, while sections of simple meter are nearly arhythmic in their freedom. Once again we see that attacks here do not solidify any sense of regular pulse–a characteristic amplified through the addition of tremolo. Measures 16-22 are built upon diminishing rhythmic values within the repeated pc11, pc8 motive. This fading away of rhythm is shown in Figure 11-6 above. Mamlok uses row-form Ie in such a way in these closing measures as to incorporate tonal references, via F and F# major [037] 3-11 trichords, presented as grace-note simultaneities. Tonal references play an important part in connecting the disparate intermezzi into a unified whole.

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Figure 11-7, Roadmap of the first intermezzo

Consecutive rows are often connected through the use of pitches from row-orders numbered 10 or 11. Figure 11-7 above shows the first intermezzo’s path through the matrix, traceable by following the arrows placed outside the matrix in numerical order. Though it may not be immediately evident how these various rows are linked, their connectedness develops over time, and row-forms eventually come to be tied together in multiple ways. For example the first pair of row-forms, P9 and It, do not have a shared pitch at their end and beginning, as will become common later in the work, as pc10 is not near either end of P9. But for the next row change, occurring at measure 5 in the contrasting 2/4 phrase, pc1 appears as a grace note marking the end of It and the beginning of RIt. That pivot note can be seen above in Example 11-2. Later, in measure 12, a dyad combines the last pitch from P8 with the first pitch of I7. The next row also overlaps similarly, on the downbeat of measure 14. This time, however, the connection is made stronger through pc10’s dual membership in both I7 as order number 11, and Pt as order number 0. (Both transitions between rows are shown below in Example 11-4.) From adjacent presentations of rows with no shared pitch at the beginning of this first intermezzo, to dyads created to tie the ends of rows together, to common pitches linking rows, Mamlok is already beginning to take advantage of common adjacencies between rows in order to trace a winding path toward the center of the matrix. This technique will continue to develop, increasing the structural tension to an eventual breaking point.

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Copyright © 1992 by C.F. Peters Corporation. Used by permission. Example 11-4, Intermezzo I, m. 12 (l), and m. 14 (r). Order numbers are shown above pitches, and to the left of simultaneities.

Finally, regarding Mamlok’s row choices in the first intermezzo, I have already noted that she explores the HRIC property within the matrix, but elements of trichordal invariance are also present in her rows. This is a natural result of the matrix’s design, but it is interesting to note when Mamlok chooses to employ these row pairs and when she chooses to avoid them. Figure 11-8 below shows the trichord invariances that exist between successive rows in the first intermezzo.

Figure 11-8, Invariances in successive rows in the first intermezzo

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Figure 11-9, Trichordal invariance pattern found throughout the first intermezzo that exists between Px to Ix+1 related rows and their inverse

The most common pattern that appears between successive rows is Px to Ix+1, or conversely Ix to Px-1, which results in the organization seen in Figure 11-9 above. Other patterns emerge at later points in the piece, playing a role in the organization of each intermezzo, as well as the work in its entirety.

Intermezzo II: In Waltztime At 18 measures, the second intermezzo is slightly shorter than the first, though more dense in texture. These two intermezzi open similarly with straightforward row presentations, but the second intermezzo focuses on R9 and RIt, retrogrades of the first intermezzo’s opening P9 and It. As a result we find a different pattern of trichordal invariance.

Figure 11-10a, Invariances found between each successive row of Intermezzo II

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Figure 11-10b, the trichordal invariance pattern found at the opening of the 2nd intermezzo that exists between Rx to RIx+1 related rows and their inverse

This intermezzo also makes use of a pivot pitch to move between rows in the most texturally dense section of the piece that falls more or less at its center, measures 6-9. Specifically, the tetrachord that falls on the downbeat of measure 8 uses pc7 as order number 3 of R8 while that pitchclass simultaneously functions as order number 11 for the following Rt. Additionally, in measure 4 Mamlok begins to take some liberties with regard to row order, switching order numbers 3 and 1 at the end of a previous use of RIt. These liberties are stretched even further in the intermezzo’s closing measures.

Figure 11-11, R9 segmentation in Intermezzo II, mm. 14-18

In the final five measures Mamlok also segments the final row form, R9, specifically to bring out consistent 3-3 harmonies, as seen in Figure 11-11 above; the use of R9, which is the same permutation that opened the intermezzo, creates a satisfying closure. Not only are some liberties once again taken in the usage of this row, but order numbers 11-8 are noticeably absent. In addition to the unique segmentation of the row occurring here are the voice leading considerations between the low voice’s quarter notes, and the upper’s half notes, where a pitch from the lower voice becomes the next half note, as shown in Example 11-4 below. In contrast to the uncomplicated manner with which pitches are deployed throughout this intermezzo, the final measures expand the process by gradually replacing only one pitch while retaining all others. A similar method of pitch

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deployment exists in the middle section of the next intermezzo, which is also the middle of the entirety of the Five Intermezzi.

Copyright © 1992 by C.F. Peters Corporation. Used by permission. Example 11-4, Intermezzo II, mm. 14-18, showing order numbers for R9 above each note and to the left of each simultaneity. Non-adjacent order numbers in mm. 15-16 are used to maintain consistent 3-3 harmony.

The closing measures of the second intermezzo also continue the basic gestural design characteristics of the whole. An alternating up-down-up attack pattern between the two voices is consistent in every measure, save for measures 9-11 where a syncopation briefly inverts the pattern. This disorienting temporal displacement foreshadows a similar attribute that will be extended significantly in the middle intermezzo. Immediately following the gestural invasion of mm. 9-11, measure 12 brings back the motive from the previous intermezzo that appeared in the contrasting, arrhythmic phrases. The motive here is taken specifically from the same formal position as that of the first intermezzo, meaning the final phrase just before a closing codetta. The ascending ic3, perhaps the most significant interval of the piece from a structural standpoint, from pc8 to pc11 is a retrograde of the motive as it appears at the close of the first intermezzo. In addition to the returning pitch content there is a tremolo on pc11 here–notably the only instance of tremolo within the 2nd intermezzo. Finally, the retrograding of this motive is consistent with this intermezzo’s design of sticking to Rx and RIx rows throughout, contrasting with the first intermezzo’s near complete avoidance of such permutations.

Copyright © 1992 by C.F. Peters Corporation. Used by permission. Example 11-5, Intermezzo II, mm. 12-13. Return of motive first presented in Intermezzo I, as seen in Example 11-3 above.

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This motivic return is noted by Mamlok herself as central to her thinking. She says, “…you find in most of my pieces that there is a return to something that…makes easier listening I think.” 3 This is made abundantly clear in the intermezzo that follows.

Figure 11-12, Roadmap of the second intermezzo

Intermezzo III: Gently Swinging A Section: mm. 1-48 Intermezzo III is the longest of the set. Its 126 measures far outnumber that of any other intermezzo in the collection. The A section here is clearly evocative of a waltz, and marked “Gently Swinging” with dotted half notes on each downbeat ringing underneath quarter notes in the upper voice on beats 2 and 3. 4 The texture and contour remain consistent throughout the outer sections of this ternary form, though the rhythmic pattern eventually shifts by a quarter note, obfuscating the waltz as it moves closer to the B section. The B section, measures 49 to 76, contrasts considerably and the processes of following rows in the matrix start to break down, with musical material informed more by the implications of the matrix, and patterns of connecting hexachords found within it, than by following rows in their entirety.

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Copyright © 1992 by C.F. Peters Corporation. Used by permission. Example 11-6, Intermezzo III, mm. 1-4. Order numbers 10 and 11 are inverted from the outset, creating overlapping transitions between successive rows.

The 3rd intermezzo’s opening use of I5 is straightforward, with order numbers presented one after the other without alteration. With R1’s appearance in the 4th measure, order number 11 from the opening row is reinterpreted as order number 10, a pattern that continues throughout the A section. These common tones link each row seamlessly, and eventually develop into entire row segments that successive permutations hold in common. This characteristic is central to Intermezzo III, pointing to the B section’s spiraling into the center of the matrix. The next rows, R1 and RI9, overlap in the same way. This backtracking to connect rows is what will allow Mamlok to traverse an unconventional path through the matrix. An increasing amount of freedom is taken with regard to pitch usage as the A section continues. In measure 19, order numbers 10 and 5 of R0 are made adjacent for the sake of having a descending minor 3rd, hinting at F minor harmony, which itself becomes a recurring theme. Additionally there is a recurrence of order number 10 moving through the row forms in this section with a fair degree of freedom, as well as the omission of pitches at either end of a row. Order numbers 6 and 0 in this instance are left out entirely. The rows that follow become gradually more un-row-like with RI0’s order number 10 missing, while order numbers 2-0 also function as order numbers 0-2 of P6, in an order number palindrome that uses pc6 as a pivot and exploits the only invariant trichord between the two rows. This is shown in Example 11-7 below.

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Copyright © 1992 by C.F. Peters Corporation. Used by permission. Example 11-7, Intermezzo III, mm. 19-25. R0 omits order number 0 and switches out order number 6 for order number 10. RI0 omits order number 10, while its concluding trichord makes use of the only trichordal invariance with P6.

The choice of rows here, in addition to the freedoms taken with their ordering, highlights a desire to further obscure the work’s location within the matrix through a high degree of pc invariance between the rows. Aside from the {6, 9, 0} trichord that overlaps here, order numbers 3, 5, 6, and 8 of rows RI0 and P6 are also invariant. This obfuscation is exacerbated by the rhythmic shift that occurs just prior, in measure 8. The first seven measures of the intermezzo project a waltz pattern with strong downbeat and weaker 2nd and 3rd beats, but in measure 8, beat 3 doubles as the downbeat leading into the next measure. This metric shift continues through measure 42, the beginning of the final phrase of the section.

Copyright © 1992 by C.F. Peters Corporation, Used by permission. Example 11-8, Intermezzo III, mm. 8-12. The metric shift that occurs on beat 3 of measure 8 is heard to function simultaneously as a downbeat–a shift that continues through nearly the entire A section.

Row overlaps only increase as the B section draws closer. Moving between RI0 and P6 in measure 24 there is an overlap of 3 pitches, followed by the significant 8-pitch overlap between P6 and I1 in measures 25-29, shown in Example 11-9 below. This leads to order numbers 0, 1, 3, and 7 of I1 acting simultaneously as order numbers 8-11 of P6. P6 is therefore drawn out, exposing 3, then 2, then 1 new pitch per measure

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from measures 26 through 29, gradually fading out one row while fading in the next. I1 is prolonged across measures 30 and 31 through a string of ascending ic3, taking off from order number 9, and taking the place of the row’s final two pitches. In this extension of the row, Mamlok uses repeated pitch content derived from the same idea that generated the matrix. This extrapolation ultimately points to the B section, where such a process is commonplace. In this case, mm. 30.2-31.2 repeat three of the final four pitches of the row that are presented, order numbers 6-8, with the addition of pc10, forming the {0} 4-27 tetrachord, which includes a 310 subset–that which the structure of the matrix is based upon.5 Adding to the growing ambiguity is Mamlok’s increased usage of the HRIC property of the matrix, in that mm. 29-33, leading up to the palindrome in the following measures, can be seen as the 2nd hexachord of R5 or the first of It–those rows being HRIC pairs.

Copyright © 1992 by C.F. Peters Corporation, Used by permission. Example 11-9, Intermezzo III, mm. 26-33.1. Note the extensive overlap between P6 and I1, and ascending ic3 chain built on {0} 4-27.

It is perhaps not possible to definitively state from exactly which rows the material in Example 11-9 above is generated. Each pair of rows has different overlapping characteristics, and various liberties are taken with regard to pitch order and omission. The impetus of this section is the idea that pitch generation is focusing in on the center of the matrix. From the outset of the first intermezzo, we have seen Mamlok’s increasing desire to showcase the invariant relations between rows, and now we will begin to see an increasing amount of extrapolation where material is not strictly associated with any specific row ordering. In the measures following the aforementioned multiple row overlaps, and beginning with the pc2 in measure 33 to pc4 in measure 34, seen in Example 11-10 below, a palindrome is formed with the pitches in measures 36-37. The pitches of measures 36-37 are order numbers 0, 1, 2, and 4 of I4, discounting order number 3’s pc5. Although pc5 does appear

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in measure 33 immediately preceding pc2, the order is reversed at either end of the palindrome. The palindromic tetrachords are separated by a 418 tetrachord that sits between them, binding their content through interweaving melodic lines. Both the 4-12 tetrachord generated by the palindrome, and the 4-18 tetrachord that separates them, are secondary harmonies of Mamlok’s row.

Copyright © 1992 by C.F. Peters Corporation, Used by permission. Example 11-10, Intermezzo III, mm. 33-37 pc palindrome. Numbers above palindrome are order numbers while those below are pc integers.

Continuing with the theme of manipulating the placement of order number 10, I4 continues after the above example to present its entire row except for order number 10, which would be pc8. Instead, pc3, order number 6, is repeated. Just as with earlier alterations, this is done to accommodate an ascending ic3 instead of what would be an ascending ic4. The final row in the A section, RI8, undergoes considerable changes by beginning with order number 9, and upon reaching order number 3 is followed by 10 and 5 before 11 appears, followed by 0, 1 and finally 2. Note that this pairing of order numbers 10 and 5 happened once before, back in measure 19. There may also be room to argue that Pt is the generating row here, as many of the pitches taken out of order in RI8 actually line up correctly under Pt. As before, there are several pitches that belong to the same order number of both rows, so that at the very least it can safely be said that Mamlok is continuing to explore different kinds of invariance between the rows. RI8 may be the best choice of row with which to label this final phrase, but the number of liberties taken with its ordering make it somewhat far removed from serial dogma. We can see how both RI8 and Pt line up with the score in Example 11-11 below.

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Copyright © 1992 by C.F. Peters Corporation, Used by permission. Example 11-11, Intermezzo III, mm. 40-48. The closing measures of the A section almost completely break away from the matrix with row ambiguity giving way to overlapping secondary harmonies.

Again, most of the changes to row order are made to accommodate ic3 motion, and as such the A section closes with a series of chords not taken from rows, but formed out of ic3 chains. Following the {2} 4-28 tetrachord that concludes the final row form, after the aforementioned manipulation, there appears a series of imbricated 3-11, 3-10 and 3-3 trichords, and finally a descending arpeggiation of a 4-17 tetrachord. When considering aberrations of the rows, and the sections of the piece that do not follow any row-forms in their entirety like the ending of the A section shown above, we find that the following sonorities appear: tetrachords 4-17, -27, and -28, and trichords 3-3, -10, and -11. All of these simultaneities appear as secondary harmonies within the row. Example 11-11 above makes clear the underlying procedure for this first section of Intermezzo III. Each of the rows connect with increasing overlap, working toward the center of the matrix. The HRIC property between rows aids in obscuring the generating row until the conclusion of the section brings the piece to the center of the matrix.

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Figure 11-13, Intermezzo III, mm. 1-48 roadmap. The gray pc integers in the center of the matrix begin with measure 42, where the generating row becomes entirely ambiguous, as shown in Example 11-11 above. Following the arrows within the gray sections, beginning with the outside layer and then the inside, traces the pitch usage from m. 42 to m. 48.

B Section: mm. 49-76 Marked with a doubling of the tempo, and a double-bar divider, the significantly-contrasting B section comprises measures 49-76. Further contrast between the sections is evident in the rhythmic design, where the outer A sections are entirely in 3/4 while the middle section now shifts to a variety of rapidly shifting compound time signatures. This section picks up where the preceding one left off, and as the strategy for the previous section was to work toward the center of the matrix, the goal of the middle of this intermezzo is to unwind itself, working back to the outer boundaries of the matrix. Not only is this the exact middle of this intermezzo, but, because of the work’s five-movement design, these measures stand at the center of the entire work. We have arrived at the conceptual nucleus of the work as well: spiraling in toward the center of the matrix and then spiraling back out again. This is something we have seen taking shape since the outset of the first intermezzo.

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Taking a step back from pitch considerations momentarily, the B section also employs a rhythmic and formal design that aids in controlling pitch generation. There are two types of measures here: those containing strictly linear content, and those containing simultaneities. These measures alternate systematically in a steadily expanding pattern until measure 77, the return of the A section. This design is such that the number of attacks in the measures of simultaneity are equal to the number of measures of linear material that preceded it plus one. For example, the first two measures of the section consist entirely of linear material, and they are followed by a measure with three attacks. Next there are three measures of linear material followed by a measure of four simultaneities, and so on.6 This pattern continues throughout the section, up to the closing two measures with 7 successive attacks. 7 As mentioned above, this pattern aids in organizing pitch content. The additive process that is at work throughout this section with regard to form is also at work with pitch in the measures of simultaneity. Looking at measure 51 in Example 11-12 below, the first measure of simultaneities after the opening two measures of linear material, we can see on the surface that beat one contains a tetrachord, beat two a pentachord, and beat three a hexachord. Comparing m. 51 to the previous two measures, we can see that the tetrachord takes all its pitches from the first linear measure that precedes it, as does the pentachord; the hexachord’s pitches, except for pc5, come from the immediately preceding measure. Measure 60 is the only exception to the rule of constant expansion in the B section. Incidentally this measure is located at nearly the exact center, creating an ascending motion that will be resolved with the next appearance of simultaneity in measures 66-67. In measures 66-67, the additive pitch process is perhaps clearest as new pitch classes are introduced with each attack. Across the seven simultaneities we have pc9 with octave doubling, followed by {59}, {059}, {5890}, {589e0}, and {7859}.8 Tracing these patterns through the section unveils that the additive process is a natural result of the unfolding that is occurring as the piece moves outward from the center of the matrix. Also, since we have arrived at the matrix’s center, complete rows only begin to appear toward the section’s end. Therefore, a measure-to-measure approach is the best way to analyze pitch usage in this section. This becomes clear when looking at Figure 11-14 below, which shows a complete measure-by-measure breakdown of the B section.

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Copyright © 1992 by C.F. Peters Corporation, Used by permission. Example 11-12, Intermezzo III, mm. 49-62. In this excerpt we can see a portion of the systematic pattern that creates the formal design of the B section. Two measures of 3/8 are followed by a measure of 9/8 containing three dotted-quarter attacks; this is followed by two measures of 3/8, and a measure of 8/16, which are followed by a measure of 12/8 containing four dotted-quarter attacks. This pattern continues to expand until reaching the concluding seven dotted-quarter attacks, which is followed by the return of the A section, which is strictly in 3/4.

Figure 11-14, Intermezzo III, mm. 49-76 roadmap. Beginning with the hexachord of P7 this section spirals out toward the edges of the matrix on a measure-tomeasure basis. Measures not appearing contain chords that draw their pitches from the measures that precede them.

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In Figure 11-14, we see Mamlok’s pitch usage as it steadily grows as a result of the processes set in motion beginning in measure 49. Some interesting things begin appearing when traveling around corners, for example looking at measure 56, in Example 11-12 on the previous page, pc2 seems to be tacked on to the end of the measure, leaping upward to prepare the descent that begins in the following measure. On the matrix we can see that in P3, order numbers 2-9 are used in measure 56 in order; pc2 is around the corner, order number 3 of I1, the row-form down which measure 57 will travel. Measures 68-69 are the first to present a full row, though we have been getting closer to this point. In measure 69 order numbers 10 and 11 are switched once again, as we have seen several times already. Despite being past the middle of the piece, liberties are still taken in pitch exposure, although pitches that are skipped over are recovered by measures that follow. For example, measure 71 has an interesting way of turning the corner of its row to fill in a gap left by measure 63, that while traveling down I6 skips over pc10, order number 10. After picking up measure 63’s pc10, measure 71 continues from where it left off, moving across R1.9 Elsewhere Mamlok continues to take advantage of the invariant properties within the matrix in order to round corners in the path, crossing between row-forms.

Copyright © 1992 by C.F. Peters Corporation, Used by permission. Example 11-13, Intermezzo III, mm. 91-93 (l) and 97-98 (r)

The return of the A section features many of the same techniques that were used in the initial A section, though a few things are worth mentioning. Two palindromes overlap with one another in measure 91 resulting in an overlapping of 3-11 trichords, and in measures 97-98 the same two 3-11 trichords, one F major and one F minor, use the F as a pivot. The same 3-11 trichord appeared earlier in the piece, and, as before, it falls outside any specific row.

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Finally, to close this pivotal intermezzo beginning in measure 115, with pc0, we see adjacent instances of pitches from each of the 4-28 tetrachords in palindromic order. With each instance of 4-28 the tetrachord is increasingly compressed, from five pitch classes to four, down to the single pitches of 4-28 {1} and 4-28 {0} that close the intermezzo. This contraction of motives contrasts and resolves the overall tendency toward expansion that drove the B section. This closing is shown in Example 1114 below.

Copyright © 1992 by C.F. Peters Corporation, Used by permission. Example 11-14, Intermezzo III, mm. 115-126. The systematic compression of 4-28 tetrachords is used to bring this central intermezzo to a close. Pitches from each of the 4 transpositions of the 4-28 tetrachord are rotated in repeating palindromic order from 4-28 {0}, {1}, {2}, {1}, {0}, {1}, {2}, {1}, and finally {0}.

Intermezzo IV: Agitato The 4th intermezzo is the only movement to use a single row-form: that which opened the work, P9. Once again the opening two order numbers are switched. After a complete presentation of the row, it is run backwards, but only as far as the first tetrachord. This is similar to a technique used in the first intermezzo where pitches were repeated to prolong the row. The tetrachord is expanded in measure 4 to include six pitches, the first hexachord of P9. It is not until measure 16 that a full exposure of P9 is brought back, again with order numbers 0 and 1 switched, as are 10 and 11. Throughout the intermezzo, the pitches become increasingly spread out, dissipating in their frequency of exposure such that entire phrases consist of only a select few repeated pitches. The intermezzo begins by exploring adjacencies, and the row as a whole, before moving to disparate

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segments that explore 4-24 whole tone tetrachords and trichords. Often in this intermezzo, with a texture like that of the 2nd, the upper voice’s pitches will be taken from the lower voice, similar to the technique that closes the 2nd intermezzo. Pitch usage in the first half of the intermezzo is shown below in Figure 11-15.

Figure 11-15, Intermezzo IV, row partitioning within the first half of the intermezzo

The remaining 12 measures continue in much the same way. P9 is presented in its entirety for the last time in measures 16-18. Closing the piece, Mamlok first uses the final pentachord from the row, immediately followed by the first hexachord. Those first six pitches of the row dissipate in much the same manner as the beginning of the intermezzo, before finally coming to a close on a 4-12 tetrachord–the same as that which opened it, in the exact same gesture. The tremolo of the final pitch of the intermezzo reminds us of the tremolo pitches that have occurred at other points in the work, notably intermezzi I and II.

Intermezzo V: Vivo

Copyright © 1992 by C.F. Peters Corporation, Used by permission. Example 11-15, Intermezzo V, mm. 1-3

The final, and shortest, intermezzo immediately takes liberties with its opening row, as seen in Example 11-15 above. The pitch classes of I8 are repeated several times, and project a descending D minor triad. This is

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followed by a two pitch class overlap between the final order numbers of I8 and the first of R3 in measure 5; R3 overlaps its final trichord with the first of I9 at the end of measure 6, taking advantage of the invariances between this trio of rows. The opening of this intermezzo is made more significant by the fact that it is the first time in the entire work that two HRIC pairs are juxtaposed in their entirety. This direct usage of the HRIC property is made even more significant in that, in taking up measures 1-6, this pair of rows comprises the entire first two phrases of the intermezzo. Following these phrases a brief contrasting idea is introduced in measures 8-9 before a return to material reminiscent of the first phrase actually closes the piece, and in turn the work as a whole.

Copyright © 1992 by C.F. Peters Corporation, Used by permission. Example 11-16, Intermezzo V, mm. 4-7, once again displaying the use of invariances between rows in order to overlap and create a smooth transition between a phrases

A final rearrangement of order numbers repeats several members of I9, while 9 and 8 are later taken out of order such that another descending 311 D minor triad is formed in measure 8, exactly as in the opening measures of the intermezzo. 10 The intermezzo closes somewhat inconclusively on the 6th eighth note of measure 13 with a 4-2 tetrachord that is to be left ringing beyond the final double bar. The final row-form, P5, is without order number 8, pc0. The use of P5 to close a work that places a certain amount of importance on the ic3/ic4 dichotomy throughout the work that began with P9, is also perhaps noteworthy.

Conclusions Ursula Mamlok’s individualistic use of serial methods in her Five Intermezzi describe a process where the path traveled through the matrix is closely tied to surface-level considerations that reach beyond pitch generation, such as elements of form and rhythm. By taking advantage of the invariant relationships that are a result of the interval structure of the

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matrix, Mamlok creates an underlying generator that functions within the matrix itself as another level of structure. Elements within the row take the shape of returning motives and cross-references that are developed, simultaneously breaking away from the matrix, while in some ways remaining tightly bound to it. In this chapter I have detailed the many ways that Ursula Mamlok used a single matrix to generate these five separate pieces that combine to create a multi-movement arch form. Mamlok’s spiraling path through the matrix confirms Joseph Straus’ observations, and the motivic recurrences, which Mamlok herself mentions as an important characteristic of her compositional thinking, combine with formal considerations in tying these works together into a cohesive unit.

Notes 1

Roxane Lise Prevost, “A Woman Composer Among Men: A Theoretical Study of Ursula Mamlok’s Serial Works” (PhD diss., State University of New York at Buffalo, 2003). 2 Joseph Straus, Twelve-Tone Music in America (Cambridge: Cambridge University Press, 2009), 141. 3 Roxane Prevost, “Conversations With Ursula Mamlok,” Ex Tempore: A Journal of Compositional and Theoretical Research in Music 11/2 (2003): 125. 4 Though Intermezzo II is marked “In Waltztime,” the tempo is seemingly the only waltz-like element present. Its up-down-up attack pattern denies a true waltz feel, and the constantly changing time signatures further prevent any semblance to a traditional waltz. Intermezzo III, on the other hand, is mostly in 3/4 and evokes a waltz through its rhythmic construction. 5 In Figure 3 I refer to the particular 4-28 tetrachord that this 3-10 subset is derived from as {0}, being that pc0 is the “root.” 6 As the pattern grows to require six and then seven simultaneities, toward the close of the section, the simultaneities are split into two measures. The rhythmic value of every simultaneity within the B section remains fixed at a dotted quarter. 7 This pattern does have one slight change toward the end where a pair of measures totaling six successive attacks is followed by seven measures of linear material, which is then followed by two measures that combine to accommodate the final seven attacks. 8 In that measure, I am speculating the reason that pc11 and pc0 are left out of the final chord is because if those pitches remained, holding the same voicing as Mamlok had done with the rest of the chords in the measure, the sonority would be impossible to play.

Exploring New Paths Through the Matrix in Mamlok
s Five Intermezzi

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Measure 74 presents, possibly, a typo, where the written pc11 appears to replace the pc10 in the matrix. Given that at this point in the section entire rows are presented as given, the flat was likely left off the score mistakenly. 10 In measure 10 of this final intermezzo, order number 9 is missing from RI5. This missing pitch would be pc1, but instead pc0 appears. This seems odd as the entire row is otherwise presented without alteration. Using pc1 would create ic4 from the preceding pc9, and ic3 from the pc10 that follows, whereas pc0 forms an ic2. As such, using pc1 would seem to be a better fit, as the entire work makes several references to the ic3/ic4 dichotomy. This possibly points to another small error in the score.

PART III: POP MUSIC, JAZZ, AND ANCIENT AND SPECULATIVE MUSIC THEORIES

CHAPTER TWELVE “LITTLE HIGH, LITTLE LOW”: HIDDEN REPETITION, LONG-RANGE CONTOUR, AND CLASSICAL FORM IN QUEEN’S BOHEMIAN RHAPSODY JACK BOSS

My first exposure to the music of the British rock band Queen came in the mid 1970s, as they were beginning to establish an international reputation, and as I was graduating from high school and preparing for my first year of music school. I remember the ambivalence I felt toward them as a typical Midwestern American teenager who was, atypically, obsessed with complex, difficult music. On the one hand, they were already well known for their “gender-bending” sexual ambiguity; this was mildly disturbing to someone who was still trying to figure out his own sexuality. On the other hand, their complex, chromatic harmonies, good voice leading, and multiplicity of vocal and guitar timbres were fascinating to me. The literature on “Bohemian Rhapsody” seems in some strange way to mirror my youthful attitude toward its creators, in that it focuses mainly on the same two topics. Excellent studies have been written by Sheila Whiteley, Judith Peraino and Ken McLeod exploring the ways in which the song represents and legitimatizes the gay experience, particularly the whole process of “coming out” and its consequences (which Freddie Mercury was living through as he wrote the song).1 But there are numerous other studies that focus on more musical and technical issues, such as the complex, time-consuming procedure of recording the song and the subtle variations in timbre that resulted. Examples of this approach include Mark Cunningham’s chapter on it in Good Vibrations, as well as two documentaries that were filmed about the song recently, “The Story of Bohemian Rhapsody” which aired on BBC Three in 2004 and “Inside the Rhapsody,” which may be found on the DVD Queen: Greatest Video Hits

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1.2 The latter documentary features Brian May, Queen’s guitarist, explaining in painstaking detail how the large choral sound in the “operatic” section was created by three band members overdubbing their own voices multiple times, how John Deacon’s bass sound and Roger Taylor’s drum set sound were composites of multiple tracks recorded with different kinds of resonance, and how his own “guitar orchestration” was enhanced by combining signals from different pickups. Still, there are other musical features of “Bohemian Rhapsody” that were captivating to me in 1975 (and still are), that the literature does not address. These include its apparent borrowing of features from classical form, changing them in interesting ways; its key scheme that seems much too well organized to be “rhapsodic,” but rather seems to express the song’s basic message somehow, and its use of motives to help the music hold together over the long span as well as to portray the meaning of the text. I will explore all three of these issues from the dual perspectives of Schenkerian and neo-Riemannian analysis. I illustrate the form and key scheme of the song in Table 12-1. The rows on this table indicate the names of the formal sections that are found in some published and online transcriptions of “Bohemian Rhapsody” and some of the literature on the song,3 parts of traditional sonata form that the sections resemble in certain ways, the key area or areas of each section, and finally, the text of each section. Boldface indicates phrases or words in the text that repeat, some of them immediately, others at places distant from one another in the song. These phrase repetitions create text motives that interact in interesting ways with the song’s musical motives, as will be seen in the following discussion.

1-15

Is this the real life? Mama, just killed a man, Is this just fantasy? Put a gun against his head, Caught in a landslide, Pulled my trigger, now he’s dead. No escape from reality. Mama, life had just begun. Open your eyes, But now I’ve gone and Look up to the skies and see. throwed it all away. I’m just a poor boy, Mama, ooh, I need no sympathy, Didn’t mean to make you cry. Because I’m easy come, If I’m not back again easy go, this time tomorrow, Little high, little low. Carry on, carry on, Anyway the wind blows, As if nothing really matters. Doesn’t really matter to me.

Text

Table 12-1, Form Chart for Queen, “Bohemian Rhapsody”

Bb major → Eb major

Bb major → Eb major

Bb major

Key area

Too late, my time has come, Sends shivers down my spine, Body’s aching all the time. Goodbye, everybody, I’ve got to go. Gotta leave you all behind and face the truth. Mama, ooh (Anyway the wind blows) I don’t want to die. Sometimes wish I’d never been born at all…..

Exposition (2nd repeat)

Exposition (1st repeat)

Intro

Verse 2

33-48

Verse 1

15-32

Equivalent in sonata form

Traditional section Intro name (in published and online transcriptions)

Measures

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Opera section

First Development

Guitar solo

Transition Eb major (ends with 6-measure transition; last 3 mm. on dominant pedal)

Second Development

Hard Rock section

90-116

Table 12-1, continued

I see a little silhouetto of a man. So you think you can stone me Scaramouche, Scaramouche, and spit in my eye? will you do the fandango? So you think you can love me Thunderbolt and lightning, and leave me to die? very, very frightening me. Oh baby, can’t do this to me baby, Galileo (4x), Galileo Figaro, Just gotta get out, Magnifico--------------------Just gotta get right out of here. But I’m just a poor boy, Nobody loves me. He’s just a poor boy from a poor family. Spare him his life from this monstrosity. Easy come, easy go, will you let me go? Bismillah! No! We will not let you go! Let him go! Bismillah! We will not let you go! Let him go! Bismillah! We will not let you go! Let me go! Will not let you go! Let me go! Will not let you go! (never, never) Let me go----------------------No, no, no, no, no, no, no! Oh mama mia, mama mia, let me go Beelzebub has a devil put aside for me.

Eb majoro A major A majoro Eb major (ends with 4-measure dominant pedal in Eb)

55-89

47-54

Oooh, Oh yeah. (2x) Nothing really matters, anyone can see. Nothing really matters, Nothing really matters, To me. (Anyway the wind blows.)

Eb majoro F major

Recapitulation (abbreviated and varied); Coda (at m. 126)

Outro

117-132

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The Introduction to the song is set in Bb major, which as we shall see, later becomes the dominant of the song’s principal tonic, Eb major, creating an auxiliary cadence. A four-part chorus (created by Freddie Mercury overdubbing his own voice) sings of the main character’s uncertainty whether his situation is real or not, followed by an injunction to “open your eyes.” Then he describes his situation—he’s just a poor boy, but we shouldn’t sympathize with him, because his condition doesn’t really matter to him—he can go either of two ways. This could signify life or death, pain or happiness, or as Whiteley and Peraino suggest, gay or straight.4 The musical setting of the Introduction is represented by Schenkerian graphs in Examples 12-1a, b and c. It incorporates three main gestures. The first is a motion through scale degrees 5, natural 4, flat 4, and 3 in the tenor in mm. 1-4. This is the first instance of a class of motives that descend through the diatonic scale with chromatic passing tones in certain places, which will become crucial components of the bass line later on. Also introduced for the first time in mm. 1-4 is the neighbor motive, which manifests as a lower neighbor over the entire top line, and six times as an upper neighbor closer to the surface. The neighbor, diatonic and chromatic, single and double, complete and incomplete, will become the central motive of this song. m. 1

2

Lower neighbor

3

4

Chromaticdiatonic descent

Upper neighbors

Bb: I

add6

7

[V V ]

Is this the real life? Is this just fantasy?

V 43

V7

Caught in a landslide, no

I

I 64

escape from reality.

Example 12-1a, “Bohemian Rhapsody,” Introduction, first gesture

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Example 12-1b illustrates the Introduction’s second gesture in mm. 511, a descending motion from scale degree 3 on “open your eyes” through 2 on “no sympathy,” transferring up an octave to 1 on “easy come.” In measure 7, scale degree 3 is ornamented by an incomplete upper neighbor Eb, covered by G and Bb, on the word “see” as the chorus is telling us to “open your eyes, look up to the skies and see.” The move up to the subdominant through a rising contour is thus associated with some sort of transcendence or vision toward a higher plane of existence. Freddie Mercury will use the key of Eb later on, after it has established itself as tonic, in the “opera” and “hard rock” sections to connote a similar idea— that of rising above prejudice and judgment. m. 5

6

7

8

9

^3

10

11

^2

^1

Chromatic double neighbor

(

)

Progression “up” to Eb Bb: vi

7

7

[V ]

IV ii7 V7

I

Open your eyes, look up to the skies and see. I’m just a poor boy, I need no sympathy, because I’m easy come, easy go, little high, little low.

Example 12-1b, “Bohemian Rhapsody,” Introduction, second gesture

At the end of the second part’s middleground 3-line in Bb, mm. 10-11, the first scale degree is ornamented by a chromatic double neighbor figure on the words “easy come, easy go, little high, little low.” This motive is especially remarkable in that it corresponds to a repeating phrase in the

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text, “easy come, easy go,” which returns later during the operatic section, set to exactly the same music (look forward to Example 12-4c). The Introduction’s third and final gesture appears in Example 12-1c. It consists of a second descent through scale degrees 5, 4 and 3, with a chromatic descent in the bass below 5 (“Anyway the wind blows doesn’t really matter”), and an unfolded tritone below 4 (“to me.”). The tritone seems like an afterthought in this context, a simple motion from fa down to ti in the inner voice. But these same pitches will later give rise to what seem to be unexplainable modulations from Eb down to A at the beginning of the operatic section, and back up to Eb midway through the same section. As they do so, they will provide a melodic contour that is this song’s most basic metaphor. m.

12

13

^5

14

15

4^

^3 ( Eb-A tritone

Diatonic-chromatic descent

Bb:

IV

I6

)

4

4

[viiio 2 ] V 3

V7

Anyway the wind blows, doesn’t really matter to me,

I to

me.

Example 12-1c, “Bohemian Rhapsody,” Introduction, third gesture

After the choral Introduction, the song progresses to a Verse that repeats, setting a different text to the same music, which to my hearing resembles a repeated exposition in a sonata form. Like a typical exposition, the Verse section modulates from the first key to a second one;

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however, instead of progressing from tonic to dominant it goes in the opposite direction—Bb to Eb, which we will come to understand as dominant to tonic by the song’s end. The first part in Bb, mm. 17-24 (and mm. 35-42 in the repeat) is shown in Example 12-2a. It consists of two motions from scale degree 3 that are interrupted, a fitting musical metaphor for the life of this poor boy that has “just begun,” but now cannot complete itself—because he either murdered another person, or did away with the straight version of himself. The latter interpretation is consonant with Whiteley’s and Peraino’s approaches to the song as representative of Freddie Mercury’s coming-out process, and also seems agreeable with the only comment he ever made about the song’s meaning—“It’s about relationships.”5 In this interpretation, Freddie can be understood as telling his girlfriend, a boutique owner named Mary Austin, that he “killed” and “threw away” the heterosexual Freddie. He begs for her forgiveness, pleading with her not to cry over his departure. Several of the important motives established in the Introduction return here in different guises. The neighbor, mostly as an incomplete upper neighbor to the main note, a kind of “sigh” motive, occurs repeatedly on the foreground in the piano accompaniment from m. 15 on, and then transfers itself to the voice on the text “life had just begun,” expressing beautifully the pathos of that idea. The motive also occurs in expanded form three times, as D-Eb in mm. 17-19, F-G over the 5-6 motion in m. 22, and Bb-A in mm. 23-24. Directly below the third of these, the descending diatonic motion with chromatic passing tones that we had heard first in mm. 1-4 takes over in the bass line, providing a crescendo to lead in to the second part of Verse 1.

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The second part of the Verse sections establishes the home key of Eb and itself divides into two gestures. We hear a middleground Urlinie transference in mm. 25-29, followed by a tonic arpeggio down to Bb. The goal note of the arpeggio is decorated by yet another incomplete upper neighbor, Cb, harmonized with a minor subdominant moving back to major tonic. In Verse 2, only the first of these gestures returns, as the guitar solo breaks in before Mercury can get to his descending Eb arpeggio. Again, the music seems to be portraying the idea that this poor boy’s life (or his relationship with Mary) is being rudely interrupted before it can complete itself. In Example 12-2b, both bass and melody are replete with motives that have already attained some sort of representative function. The complete upper neighbor G-Ab-G in mm. 25-26 is followed immediately by an incomplete upper neighbor Ab in m. 27 that progresses down to scale degree 2, and Ab-G comes back again in mm. 28-29 as a middleground motive covering scale degree 1. At the same time, the diatonic descent with some chromatic passing tones returns in the bass at mm. 27-28 as the singer apologizes for making “Mama” cry. That returning motive is bookended by two instances of a diatonic descent from do through ti to la in mm. 25-26 and 29-30, harmonized by I, V6, and vi. This bass line and chord progression will again be paired with the diatonic-chromatic descent in the “guitar solo” section, and it returns by itself prominently at the beginning of what I am calling the “Recapitulation” section. So far, Freddie Mercury has created continuity in his song using some of the same formal and motivic techniques as the classical masters. He brings back the same harmonic progressions and melodic motives from section to section, combining them together in new ways constantly as the music progresses, a pop-music equivalent to “developing variation.” As Brian May takes over from him in the following section, he continues the developmental process, placing upper neighbors in the guitar against the do-ti-la and diatonic-chromatic descents in the bass. In this regard, it is curious that when May describes his solo in the documentaries, he characterizes it as “something completely different.”6 In fact, the basic elements in guitar and bass are essentially the same as what has come before, but May does use a different upper neighbor from the ones favored in the Introduction and Verses. This is Bb-C-Bb, which he ornaments with foreground diminutions we haven’t heard yet, including a heavy dependence on octave coupling.

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Example 12-3 provides a graph of the “guitar solo” section. Its bass line is an extended repeat of the second parts of the Verse sections, consisting of do-ti-la in mm. 47-48, a chromatic descent from re down to ti in m. 49 skipping down to sol in m. 50, do-ti-la again in mm. 51-52 and then a chromatic descent from re starting in m. 53, which reaches further down past ti, through Db all the way down to A, the tonic of the “Opera” section, in m. 55. The diatonic-chromatic descent thus grows into something else here by filling in completely with chromatic motion, and uncoiling itself to the point where it introduces what seems like a completely foreign key—A major. Over the top of this familiar material, the lead guitar soars from Bb3 to Bb4 in m. 47, then presents its first neighbor C5 in m. 48, ornamenting it with a skip up to F5 and back in 48-49 as the chord progression arrives at ii. As the bass makes its first chromatic descent in m. 49, the guitar traces a stepwise path from F5 up to the same neighbor an octave higher, C6. It resolves back to Bb5 as the chord progression moves to V7 in m. 50. Then it couples Bb5 with Bb4 through descending and ascending Bb Mixolydian scales over V7, I and V6 in mm. 50-51, reaching up to C6 for a second time on the downbeat of 52 when the chords reach vi7 the second time. This C6 is coupled to C5, followed by a repeated consonant skip figure to Eb5. Eb5 in turn then moves to its own upper neighbor, F5 in m. 53, which couples with its lower octave, F4. This final upper neighbor cannot return to Eb, however, because the bass has dropped from D3 to Db3 on the downbeat of m. 54, and continues down to Bb2, while the inner voices hold on to Db major. So the guitar arpeggiates from F5 through Ab4 to Db5 instead at the end of m. 53 and beginning of m. 54, finishing with an octave coupling down to Db4. That note is then enharmonically reinterpreted to become the third of the “opera” section’s A major triad. The literature on “Bohemian Rhapsody” interprets the “opera section” as a kind of bizarre, otherworldly trial, set in hell, perhaps.7 The man condemned of murder (or of outing himself, depending on the preferred interpretation) pleads to be let go. One chorus (representing the prosecution) responds negatively: “Bismillah (in the name of Allah)! We will not let you go.” Another chorus (the defense) pleads, in the highest register, for his release. The “opera section” is illustrated in Examples 124a, b, c and d—my discussion will locate tonal and motivic features in the section by their measure numbers, given at the tops of the four examples.

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The “opera” section’s tonality, as I described earlier, begins in A Major, a foreign key to Eb, but returns to Eb in m. 61, as the defendant begins to plead his case. Now, since Eb has already been established in the song’s Introduction as representing “rising above,” and A major is led into by a long chromatic descent in mm. 53-55, it is possible to understand the “opera” section as creating a large contour that dips down at the beginning and then rises up again. In this way, it is similar to a piece that tells another story about descent and rising up, the “Crucifixus” from Bach’s B Minor Mass. Bach’s chorus begins in E minor, attempts to modulate to A minor, down a fifth, a little over halfway through (right after the repeat of the text “crucifixus etiam pro nobis” begins, at mm. 38-41), creeps back up to E minor at m. 49, and then cadences in G major, even higher, at the end—in what I think is an attempt to portray Christ’s burial and impending resurrection through its large contour. “Bohemian Rhapsody” begins in Eb major in the two Verses (after its auxiliary cadence), descends to A major at the beginning of its hellish trial, returns up to Eb when the “poor boy” begins to defend himself, stays there as he gives his defiant response in the subsequent “hard rock” section, and then after he moves on to his final expression of indifference, rises to F major. The contour is essentially the same as Bach’s, but the main interval is different—a tritone rather than a perfect fourth. However, you will recall (if you have a quick look back at Example 12-1c) that Mercury has already prepared this same tritone, Eb to A, by the final cadence of the Introduction. The “opera” section has other features that tie it to the preceding music. Indeed, this section, more than any other in the song, is full of textual and musical motives that recur in different contexts and setting different words from their original statements. First and foremost is the neighbor motive. We hear the Introduction’s chromatic double neighbor in mm. 62-63 setting a different text motive from its original statement (“I’m just a poor boy”) and in mm. 69-70 setting its original text motive (“easy come, easy go”). A double neighbor figure derived from it that creates the harmonic progression IV6/4 - I - cto7 – I appears twice, at the beginning of the “opera” section in mm. 55-56 and in mm. 64-65. Single upper neighbor figures also play a significant role. In mm. 59-60, we hear no less than five of them on the repeated “Galileos” followed by a scale down from do to sol on “Figaro.” Then, in mm. 82-83, the same melodic figure returns, repeated three times on “Mama mia,” again followed by a descending fourth (this time from sol down to re).

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The proliferation of diatonic and chromatic neighbors in this section is one of the two main features that causes it to resemble a sonata development. (The other is the four-measure dominant pedal at mm. 8689.) The “opera” section takes melodic fragments from the introduction and exposition, brings them back in new contexts, and transforms them to create new shapes. Like the development sections of the classic masters, Mercury’s first development also expands these motives to create hidden repetitions. The most obvious examples are the repeated “will not let you go’s” in mm. 74-78, which decorate the upper neighbor F with ascending and descending consonant skips to Ab and the first and last Ebs with consonant skips to Bb a fourth below (“Bismillah”) and Eb an octave higher (“Let him go”). Other motives also come back in new contexts in the “opera” section, and lead into new motives that are in turn developed further. The chromatic bass descent that led into the first development in mm. 53-55 returns only a few measures later in mm. 58-59 to set the words “Thunderbolt and lightning.” This descending bass line leads into a 3-2-1 descent that is harmonized by an unusual, chromatic mediant-infused chord progression—bIV6/4 -[V/bIV] – major III6/4 – V - I. As Example 12-4a shows, the progression can be understood more easily in terms of neo-Riemannian transformations on its underlying triads. From this perspective, it goes through a circular motion from A major (indicated on my Tonnetz fragment as Bbb) back to A major, depending mostly on combined leading-tone and parallel transformations, but including one each of Lewin’s “subdominant” and “dominant.” Similar progressions recur, without the chromatic bass descent, at mm. 80-81 (“No, no, no, no”) and 84-88 (“Beelzebub has a devil put aside for me”), both shown in Example 12-4d.

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Examples 12-5a and b illustrate the “hard rock” or “head-banging” section of “Bohemian Rhapsody.” (The latter designation refers to this section’s appearance in the 1992 American comedy film, Wayne’s World.) My examples and form chart refer to this part as a second development section, which I think is justified because of the dominant pedal that ends the section, mm. 114-116, and the preceding measures, mm. 107-113, which gradually take on a number of musical characteristics that signal a big return—sustained chords leading to a dominant, rising melodic and bass lines, increasing numbers and intensity of accents, and finally a large ritard in m. 116. Before mm. 107-116, the first half of the “hard rock” section, Example 12-5a, presents familiar elements in an unfamiliar context. As mentioned already, the text represents the “poor boy” rising up against the judgment and/or prejudice he is facing—“can’t do this to me, baby.” The music, in its rhythm, tempo, instrumentation, and general character, takes a completely different turn from the immediately-previous measures: toward a more typical 1970s fast dance style after the Gilbert and Sullivan-like qualities of the “opera” section. But with respect to motive and contrapuntal structure, every element refers back to previous music, giving “Bohemian Rhapsody” a coherence that transcends its abrupt style changes. The Schenkerian graph of mm. 90107 shows a middleground interruption whose second scale degree corresponds with the endpoint of the question, “So you think you can stone me and spit in my eye?” As the poor boy responds in his concluding statement of the section, “Just gotta get right out of here,” the Urlinie transference completes its journey to 1. In previous music (the Verses and “opera” section), interruptions had functioned to signify the meaning of the text—the idea that the poor boy can’t complete his life or relationship. Here, however, the interruption seems to portray a more structural aspect of the text—it sets the question mark. The motives in the first half of the “hard rock” section bring together certain elements from the guitar solo with the upper neighbor, which we have seen to play an important role in every section of the song. Lead guitar and bass begin (after a falling minor sixth Eb-G) with an ascending scale fragment, recalling the rising and falling scales of the guitar solo. The scale is Eb major this time, rather than Bb mixolydian, but after the second rising scale fragment at the end of m. 91, we hear Bb-C-Bb, the same neighbor that was so prominent in the guitar solo. As both examples 12-5a and b show, both the rising scale fragment and upper neighbor continue to be featured after the voice comes in at m. 94, and all the way through to the end of the section at m. 116.

upper neighbors

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The “outro” section, mm. 117-126, fulfills the function of a recapitulation in several ways, but one must admit that this recap is a severely abbreviated and varied one. (See Example 12-6.) The recognizable musical features that return are three: the do-ti-la bass harmonized by I - V6 - vi, the incomplete upper neighbor harmonized by IV, and the upper neighbor Cb-Bb harmonized by minor iv. The first of these features comes at m. 117-18 in the recap, during the choral “oohs.” It recalls similar passages in Verses 1 and 2, as well as the guitar solo. The second feature creates a plagal cadence for the guitarist at m. 121, and recalls passages in the introduction like the first appearance of the words “Anyway the wind blows” in Example 12-1c. The third feature brought back in the outro makes the most convincing connection with the Verse sections, I think, because it recapitulates text as well as music. The source passage in the first Verse is mm. 30-31 in example 12-2b: the words “nothing really matters” were set there to a middleground Cb-Bb neighbor, of which the first note was decorated by foreground Db upper neighbors. The chord progression there was minor iv – I in Eb. In the recap at mm. 122-125, the phrase “nothing really matters” repeats three times, and its music can be understood as a compound melodic structure sustaining the primary tone G in the upper voice and projecting a Bb in the alto for mm. 122-123, rising to Cb in m. 124, then falling back to Bb (transferred to the bass) in m. 125. Again, Cb is harmonized by minor iv. The Bb to which it falls back down supports scale degree 2 of the piece’s background 3-line, and I under 1 follows in m. 126, as Mercury makes his final cadence in Eb. The final seven measures, mm. 126-132, constitute a codetta section, and are illustrated in Example 12-7. I characterized them earlier as establishing the home key of Eb, and then rising above it to F major. The way F is approached here is fascinating to me, because it pulls together two motives from earlier in the song, in the bass from mm. 127 to 129. The do-ti-la motive harmonized by I, V6 and vi forms the melodic and harmonic outline of this passage (except that minor vi is turned into a C dominant seventh chord to carry out the modulation). At the same time, this motto of the verses, guitar solo and recapitulation is ornamented with te in the bass in the second half of m. 128, turning Eb: major V into minor v, and creating a pivot chord which can also be interpreted as minor iv in F. This chromatic passing tone Db recalls the descending chromatic bass lines from earlier in the song, especially the one at the end of the guitar solo that led the “poor boy” into the foreign, “hellish” key of A major through Db as a midway point (measure 54 in Example 12-3).

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It could be possible that what we hear at the end of this song is a musical metaphor for the “poor boy” gaining a hard-fought victory over prejudice. He rises above not only the judgment brought on by his decision (going from A major back up to Eb), but also conquers those forces that would drag him down into judgment in the first place (the chromatic descent, which originally had led down to A major). He takes hold of the chromatic descent and uses it to turn the music up toward his own key of F major, in which he can proclaim his indifference over the whole matter. In this chapter, I have been able to show that “Bohemian Rhapsody” recreates quite a few of the conventions of common-practice period classical music, and changes them in interesting ways. It adapts sonata form by moving from dominant to tonic in the exposition, presenting two development sections, and abbreviating the recapitulation. Its large key scheme expands an interval that was introduced as a motive near the end of the Introduction. It falls from Eb to A, rises again to Eb, then takes another step up to F major, a contour that portrays quite well a fall into judgment and the struggle to rise above it. Finally, it uses motives, particularly the upper neighbor, to give itself coherence from section to section, fulfill the functions of the two development sections, and depict various aspects of the text through association. “Bohemian Rhapsody” was a piece that captured the attention of this young musician 40 years ago, and I believe it is still worthy of our study.

Notes 1

See Judith A. Peraino, Listening to the Sirens: Musical Technologies of Queer Identity from Homer to Hedwig (Berkeley and Los Angeles: University of California Press, 2006); Sheila Whiteley, “Popular Music and the Dynamics of Desire,” in Queering the Popular Pitch, ed. Sheila Whiteley and Jennifer Rycenga (New York and London: Routledge, 2006), 249-62; and Ken McLeod, “Bohemian Rhapsodies: Operatic Influences on Rock Music,” Popular Music 20/2 (May 2001): 189-203. 2 Mark Cunningham, “Just One More Galileo,” chapter 7 of Good Vibrations: A History of Record Production (London: Sanctuary, 1998); The Story of Bohemian Rhapsody, television documentary produced and directed by Carl Johnston, narrated by Richard E. Grant, aired by BBC Three on December 4, 2004; “Inside the Rhapsody,” in Queen: Greatest Video Hits 1, DVD produced by Simon Lupton and Rhys Thomas (EMI/Universal Music Distribution, 2002). 3 For example, the version for SATB choir and piano, arranged by Filip Tailor, which I consulted, adapted and revised as I created my own transcription for analysis. A PDF is available at http://www.chds.cz/wp-content/plugins/downloadsmanager/upload/Bohemian-rhapsody_choir.pdf (Accessed June 26, 2015).

“Little High, Little Low”: Queen’s Bohemian Rhapsody

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In Judith Peraino’s words (Listening to the Sirens, 232): If there is a lesson to be learned in this rite of passage, it is encrypted in the mischievous phrases we hear in the coda, which returns to the soft melancholic music of the opening aria: “Nothing really matters,” “Any way the wind blows.” Despite the song’s outright goofiness, these statements add a level of complex resistance to the song’s already charming subversion of macho rock and roll. Indeed, this resistance is its Bildung. I certainly heard it that way in high school. These words “brought me out,” convinced me of the cosmic triviality of my concerns about being homosexual. The song delivered a slightly nihilistic but certainly devilmay-care message apropos to my internal prosecution, defense, and mental brinkmanship. But more importantly, the song suggests resistance through the adoption of a “bohemian” stance toward identity, which involves a necessarily changeable self-definition (“any way the wind blows”). 5 Sheila Whiteley, in “Popular Music and the Dynamics of Desire,” 252-53, interprets the music and lyrics of “Bohemian Rhapsody” as an expression of the tension Freddie Mercury was feeling as he wavered between the security of his life with Mary Austin, and wanting to break away into a relationship with his first gay lover, David Minns. Mercury’s enigmatic quotation, “It’s about relationships,” is discussed in paragraphs 7-9 of David Chiu, “Unconventional Queen Hit Still Rocks After 30 Years,” The New York Times Online, 27 December 2005 (http://www.nytimes.com/2005/12/27/arts/music/27quee.html?ex=1293339600and en=5825caa9f4db1fb0andei=5090&adxnnl=1&adxnnlx=1370382818QGRh1ZgUFIQu+sNVjxW+4g, accessed June 4, 2013). 6 May’s comments can be found at 12.33-13.05 of “Inside the Rhapsody,” Part 3. This video is accessible through the official Queen YouTube channel at https://www.youtube.com/watch?v=v15oIktGJOo (Accessed June 27, 2015). 7 Ken MacLeod, for example, describes the opera section as “an Orpheus-like descent into the insanity of the underworld complete with ‘thunderbolts and lightning, very very fright'ning…’.” He goes on to explain how the section evokes “the exotic insanity of [an] underworld trial.” See “Bohemian Rhapsodies: Operatic Influences on Rock Music,” 192-94.

CHAPTER THIRTEEN SCHENKERIAN VERSUS SALZERIAN ANALYSIS OF JAZZ RICH PELLEGRIN Bebop is sometimes referred to as the lingua franca of jazz, or as the common-practice period of jazz.1 It is generally accepted by jazz musicians, educators, and scholars that “modern jazz” begins with bebop. Most of the various jazz styles to follow use bebop as their fundamental language, and aspiring jazz musicians are expected to gain fluency in it.2 It thus is natural that the work of the late Steve Larson—the scholar most associated with the Schenkerian analysis of jazz—centers on modern jazz, as opposed to what he refers to as “post-modern jazz.”3 Generally speaking, Larson treats the performances he analyzes as theme-and-variations forms. The themes involved are standards (or “jazz standards”), and are amenable to the Schenkerian approach.4 Each chorus of improvisation is essentially analyzed as a variation upon the theme, or on the “underlying voice-leading strands of the theme,” and thus a traditional Schenkerian analysis of the improvisation works relatively well.5 One limitation of this approach is the fact that the improvisations of modern jazz are often markedly unrelated to the original theme, in contrast with the practices of some prior and/or more commercialized idioms. Bebop musicians tended to view the original composition as a harmonic “vehicle” for improvisation.6 Furthermore, the melodies of popular songs, references to which would have been easily perceived, were often replaced with rapid and improvisatory bop melodies; this practice allowed the musicians to avoid copyright issues, yet still improvise over harmonic progressions that they knew well.7 The musicians Larson examines are somewhat unusual in the degree and directness with which they engage the original theme in their improvisations.8 Nevertheless, Larson’s work is a powerful testament to the large-scale organization and unity that can be achieved through thematic reference. Moreover, thematic reference is very much a matter of degree. It might

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even be said that the extent to which thematic reference is present is the extent to which an improvisation will be amenable to a Larsonian strict Schenkerian analytical treatment of it as a theme-and-variations form.9 The issue of whether or not to use a strict Schenkerian approach in analyzing jazz is highly significant, and debate over this matter has recently been renewed with the sudden passing of both Larson and Steven Strunk, as well as with the re-publication of Larson’s 1987 dissertation, with minor revisions, as Analyzing Jazz: A Schenkerian Approach.10 This question is fraught with political import: if Schenkerian analysis has been used by some as an aesthetic litmus test for greatness, then jazz scholars have a clear interest in demonstrating that jazz performances are capable of exhibiting the same sort of organic coherence that Schenker revered in European concert music. Larson has in fact demonstrated this, and in so doing has implicitly defined what might be referred to as the Schenkerian jazz repertoire.11 However, as in classical music, problems arise if one attempts to apply a strict Schenkerian approach to music that falls outside of certain boundaries, in this case, those delimited by Larson’s work; these boundaries may be clearly defined in theory, but in practice there will be a multitude of gray areas; and the repertoire amenable to a strict Schenkerian approach only represents a small portion of jazz. This means that a less strict—i.e. Salzerian—approach to much jazz is necessary if one wishes to examine it from a “structural” perspective. If the Salzerian tradition were held in higher esteem, as I have elsewhere argued that it should be, then jazz analysts would perhaps feel less as if they needed to fit the music into a strict Schenkerian model in order to demonstrate the value of jazz and jazz theory.12 Discourse pertaining to structural analysis often involves the issue of salience, whether explicitly or implicitly, and here I would like to pause and provide some background on this subject. The concept of salience covers a wide range of parameters, which may be broadly categorized as follows: first, those that are commonly studied using hierarchical levels, such as meter and grouping; second, those that are less frequently (or less independently) examined in this fashion, such as register, timbre, and dynamics; and third, parallelism (or repetition). The use of the word “salience” in this context came to the fore with the publication in 1983 of Fred Lerdahl and Ray Jackendoff’s seminal work, A Generative Theory of Tonal Music (GTTM). They distinguish salience from stability, which is more directly related to pitch space and tonal closure. In my work on Salzer, I discuss the fact that he often relied more upon salience than stability in making reductive decisions, particularly in the

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analysis of post-Schenkerian repertoire. However, he did so in an inconsistent and relatively intuitive manner. Lerdahl’s work on this repertoire, which is presented most comprehensively in Tonal Pitch Space, is in many respects a formalization of Salzerian analysis, just as GTTM is essentially a formalization of Schenkerian analysis. One of the fundamental points Lerdahl makes is that while “stability far outweighs salience in making reductional choices in diatonic tonal contexts,” salience becomes increasingly important as less tonal repertoire is encountered.13 This is consistent with Salzer’s approach. Before proceeding, it will also be instructive to consider some of the ways in which post-Schenkerian jazz practices may differ from those of common-practice jazz. This list, given in Table 13-1 below, is by no means comprehensive, but is intended to illustrate the need for jazz analysis in the Salzerian tradition. Points 8, 9, and 11 will now be further explored, as they are particularly relevant to the issue of salience versus stability. One of the fundamental premises of Larson’s work concerns the issue of pitch stability vis-à-vis the presence of ninths, elevenths, and thirteenths.14 In “Schenkerian Analysis of Modern Jazz: Questions about Method,” Larson writes: “So-called ninths, elevenths, and thirteenths occur in both repertories [classical and jazz]. And in either case, the functions of upper chord tones—including the seventh—are best explained in terms of their melodic relationships with more stable notes at more basic structural levels.”15 (As evidence that Schenkerians use the same reasoning within the context of classical repertoire, Larson cites Aldwell and Schachter’s analysis of a Ravel passage as well as analyses of a late Brahms work by Cadwallader, Salzer, and by Forte and Gilbert.) Larson illustrates this principle with several examples based upon Strunk’s work. One shows how tension tones are often heard in terms of their resolutions, sometimes creating chains of suspensions (with the harmony changing at each point of resolution); another demonstrates how a tension tone in one chord may be a stable tone in the local key, and thus may become a stable chord tone if sustained into a cadence or structurally superior harmony. Larson also acknowledges that sometimes tension tones are more coloristic in function, and lists this as being among several features of modern jazz that are anomalous in terms of Schenkerian analytical explanations.16 However, such features, which become more and more prevalent as one moves away from the common-practice period in jazz, may often be explained from a Salzerian perspective, and that is what I will demonstrate in the examples that follow.17

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1

Increased use of non-functional chord progressions in the compositions themselves, oftentimes stemming from a more modal conception (chord changes may result from a change of bass within one referential collection).18

2

Increased freedom with respect to the improvised lines played over the original chord changes (more “outside” playing—playing notes outside of the scales that naturally correspond to the chords).

3

Increased freedom with respect to the original chords themselves, on the part of the ensemble as a whole.19

4

A decrease in thematic paraphrase and/or reference.20

5

Increased tendency towards longer improvisations (partly deriving from the increased prevalence of the long-playing record).

6

Increased tendency for improvisations with a long structural arc as opposed to more even improvisations structured on a chorus-by-chorus basis.

7

Increased tendency for metric dissonance, up to the hypermetric level of complete choruses themselves.21

8

Increased emphasis upon chordal extensions (tension tones).

9

Increased use of chords with chromatic alterations.

10

In connection with points 8 and 9, decreased organization around conventional guide-tone lines.

11

In connection with point 8, decreased emphasis (by any performers, including bassists) upon more basic chord tones, often resulting in extended passages where roots are merely implied, or passages where roots may be found but are neither salient nor in the lower register.

Table 13-1, some differences between post-Schenkerian jazz practices and those of common-practice jazz.

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Figure 13-1 gives the second half of the first chorus of a 1963 Bill Evans improvisation on Victor Young’s “Stella by Starlight.”22 For much of the first chorus, Evans plays relatively traditional bop lines over the chord changes, while referencing the melody of “Stella” clearly and with some frequency.

Figure 13-1, a transcription of the second half of the first chorus of Bill Evans’s improvisation on “Stella by Starlight.” The top stave provides the original melody of “Stella.” In mm. 145–51, diagonal lines connect Evans’s improvisation with tones of the original melody. Thematic reference in mm. 152–60 is addressed separately, in Figure 13-3.23

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It is apparent that Evans has deeply internalized this theme. Listeners must also be thoroughly acquainted with the theme if they are to fully perceive Evans’s artful referencing of it. A lead sheet for “Stella” is given in Figure 13-2.

Figure 13-2, A lead sheet for “Stella by Starlight.” Bassist Chuck Israels plays the theme in this performance. Measures 17–32 correspond to the excerpt given in Figure 13-1.24

During mm. 17–24 of “Stella” (145–52 in Figure 13-1), the melody itself features much use of dissonance, but is still fully compatible with the Strunk/Larson model. The salient melodic tones of these measures are chordal extensions and/or alterations. These tension tones are sustained for several beats, but they eventually resolve downwards by step to either the root, third, or fifth of the chord. Evans’s thematic referencing is particularly evident during these measures, as indicated by the diagonal lines. An exception seems to occur in mm. 149–50, where the melody note D occurs in the middle of an arpeggio. Nevertheless, Evans emphasizes the second D by placing it on a downbeat, accenting it, and changing the rhythm of the gesture. The C of m. 151, although not emphasized much in terms of volume, articulation, or metric placement, is the highest pitch played during the B harmony; in any case, the A which precedes the C is also a tone from the original melody. More significant, however, is the way in which the C is connected to the measures that follow. During mm. 152–58 (the second half of Figure 13-1), Evans plays several ascending foreground lines, generally emphasizing the final, uppermost note of each line with both articulation and a new chord in

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Figure 13-3, an ascending middleground line in mm. 152–59 of Evans’s performance of “Stella by Starlight.” Tones from the original theme are circled.

the left hand (most of the chord changes in these measures are anticipated by one beat or half of a beat). These endpoints themselves form an ascending middle-ground structure, expressed in Figure 13-3 as a rhythmic reduction.25 The relationship of this ascending middleground line to the original theme, the notes of which have been circled in Figure 13-3, is striking. Evans uses the melody note C from m. 151 as a starting point (the C shown at the beginning of Figure 13-3 is carried over from the previous measure).26 He then departs from the original melody until the rising line naturally reconnects with it in m. 156. The return to thematic reference is more noticeable in m. 157, as Evans places both the new chord and the local-level apex tone (Gb) on the downbeat, breaking the pattern of displacement, and also plays these notes with greater emphasis. Yet in m. 158, after an expressive pause, he pushes the sequence still higher— beyond the Gb of the theme—only to then reverse course just in time to resolve the Gb down to F, in accordance with the original melody.27 Because of the way in which Evans frames this sequential passage in terms of thematic reference, these measures offer a powerful illustration of why the Strunk/Larson model works, and of the remarkable control that Evans had over his ideas. At the same time, this passage demonstrates how salience can override stability, in this case prefiguring later developments in Evans’s style (more below). For although these measures taken as a whole fit into Strunk and Larson’s model, the middleground line itself is at odds with their explanation of dissonances. While the line itself ascends, the harmony descends through the circle of fifths. Figure 13-4 shows a diagram, also featuring descending-fifths harmonic motion, that Larson uses to demonstrate how ninths and thirteenths may be viewed linearly as chains of suspensions, and thus as displacements of lower chord tones. (As regards an eleventh (unaltered)

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Figure 13-4, a generative/reductive diagram used by Larson. From Larson, “Schenkerian Analysis of Modern Jazz: Questions about Method,” 216.

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in a descending-fifths progression, if sustained into the next harmony it becomes the root, thus losing its “‘need’ to resolve.”)28 Larson explains: Viewing the model’s levels starting with level a and progressing to e, it may be seen as a means of generating ninths and thirteenths through the delay of pitches of a 5-8 linear intervallic pattern. Viewed from level e back through a, it may be seen as a means of analytically reducing such ninths and thirteenths to more stable intervals at deeper structural levels.29

The similarity between the harmonic progression in Larson’s diagram and these measures of “Stella” suggests a way to consider the Strunk/Larson model in the context of this excerpt. Taking level e (and c) from Larson’s diagram as a starting point, Figure 13-5 displays chord tones and resolutions—potential voice-leading strands—in mm. 24–31. (Measures 25–28 may be compared directly to Larson’s diagram, as the chord roots are identical.) For the sake of visual organization, the chord tones in every odd-numbered measure (the first measure of each ii-V progression) have been arranged as stacks of thirds, with the lower chord tones on the middle staff and the upper chord tones (extensions) on the upper staff.

Figure 13-5, chord tones and resolutions in mm. 24–31 of Young’s “Stella by Starlight.”30

This diagram differs from Larson’s in that he numbers tones above the bass according to the conventions of figured bass and/or linear intervallic patterns. In this and other examples, he mostly uses the numerals 5 through 10, emphasizing the linear derivation of the upper chord tones. Since Larson has already made this demonstration, I have chosen to use standard jazz designations for chord tones.

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Other adaptations have been made as well, due to the different chord qualities present in this excerpt. For example, the strand that alternates ninths with raised fifths (the bottom of the top stave) is essentially a combination of, or the halfway point between, Larson’s level c and level e. The ninths do not fall to fifths and then become ninths again, as in level c; nor do they become thirteenths and then fall to ninths again, as in level e. Rather, the ninths fall by chromatic step to become raised fifths and then fall by chromatic step to become ninths again. (This is because neither the natural thirteenth nor the natural fifth occurs in the seventh mode of melodic minor that Evans plays on the dominant chords in these measures.)31 Of course, this arrangement of tones is not intended to illustrate actual chord voicings, but rather to conceptualize some of the possible voiceleading strands suggested by these chord changes. (Other resolutions are possible; for example, the flatted fifths of the half-diminished chords could resolve to the roots of the dominant chords, forming a descending chromatic line.) These strands may then serve as guide-tone lines (middleground structures) around which melodic lines may be organized, or they may be combined to form successions of chord voicings. Indeed, one can see in Figure 13-1 that the notes of Evans’s left hand in mm. 154– 56 consist of an aggregate of three to four of the voice-leading strands shown in mm. 26–28 of Figure 13-5. His right-hand melodic line, however, follows a different path through the chord changes, one unrelated to these voice-leading strands. Any of the tones of the middleground line shown in Figure 13-3 may be resolved down by step to a root, third, or fifth in the next chord.32 Thus, at any point in mm. 152–59, Evans could have changed the direction of the line and connected with the matrix of voice-leading pathways presented in Figure 13-5. Instead, he increases the tension by continually pushing the line higher, waiting to descend until reconnecting with the melody and until the melody itself then descends.33 The Schenkerian approach often requires one to hear salient events in terms of the underlying pitch stability. In later repertoire, our perception of what is stable increasingly has to compete with our perception of what is salient. Ultimately the surface becomes less “illusory,” at least in the sense that salient events are more directly involved with large-scale structural organization. In this excerpt, we hear the tension tones of Evans’s ascending middleground line in terms of the voice-leading strands of Figure 13-5, yet we also hear the rising line itself, which, strictly speaking, works against the implied resolutions of each of these tones (as well as the descending nature of the harmonic motion overall).

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The proportion in which salience and stability are perceived in this excerpt, relative to one another, depends in part upon the level of experience the listener possesses, especially in the jazz idiom, as well as the extent of their familiarity with “Stella” and with this specific performance of it. However, it would seem that salience outweighs stability at the level of the ascending line itself, as the line is so forceful and all of its constituent tones are extensions and/or dissonant. By contrast, stability overrides salience at a deeper structural level, as we hear the ascending line within the context of thematic reference and cadential closure. The main purpose of this example has been to demonstrate the way that issues of salience and stability play out in the jazz idiom, and to dramatize the way in which the relationship between the two can change, even within one improvisation. Over the course of these sixteen measures, Evans’s improvisation moves from more stable structures organized around the original theme to less stable structures that initially do not relate to the theme. In a sense, this motion parallels the overall trajectory of Evans’s career. Readers familiar with his work will know that the texture of mm. 152–58, with its tension-filled rising line and urgent chordal accompaniment, is only a foretaste of the direction that Evans took late in his career.34 While his work in general tends to be rather strict in its conception and execution, his playing near the end of his life featured, among other things, more freedom in terms of line and (in some respects) dissonance, faster note values, bigger chords, more “rubato” and other tempo fluctuations (often pushing the tempo to an extreme degree), greater use of the outer ranges of the piano, and more pedal; in short, his playing was more late-romantic.35 More significant for our purposes, Evans’s late style is also characterized by an abundance of rising middleground structures such as that discussed above. However, compared with the “Stella” excerpt, these structures are often less repetitive (sequential), push higher, have secondary apexes, and unfold over a longer period of time. In addition, these strongly directional motions are not always entirely stepwise, and are not always as disciplined in the way that they conclude. The sense of urgency this creates is not unlike, for example, that of a continually rising Wagnerian line that seems to defy resolution every step of the way (the difference is that in most jazz, the underlying chord changes repeat every chorus—though there may be substitutions and other variations— so we hear the counterpoint of the melodic line against the root motion, even if roots are not actually present).

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For other artists, the move away from common-practice techniques took different forms. The proportion in which we hear salience and stability relative to one another depends upon the various contexts created, which are largely a function of the eleven points listed in Table 13-1. These contexts may shift rapidly during the course of an improvisation, and may involve more variables than the Evans example above. Consider, for example, Morgan’s description of Hancock’s use of superimposition in playing standards with Davis’s quintet: The types of superimposition in Hancock’s improvisations—side slips, superimpositions of bop formulas, superimposed quotation, and superimposed sequences of fragments—are woven seamlessly through the fabric of thematic paraphrase and inside bop formulas, with meticulous attention to voice-leading in both inside and outside material.36

Curiously, Morgan’s explanation of Hancock’s use of sequences is similar to what we observed above in Evans’s sequential passage, except that Evans stays inside the changes: Frequently, Hancock’s melodic sequences begin with a pattern that, as initially stated, is consonant with the underlying changes and even emphasizes thematic pitches. Subsequent transpositions of the model stray from the underlying progression, until the end of the sequence returns the line to inside material and its thematic goal pitch.37

Both Evans and Hancock begin and end with thematic reference. However, in between these structural bookends, Evans uses extensions and/or alterations in a way that works against the natural resolutions of the progression, whereas Hancock goes a step further and plays outside the changes. For Evans was an inside player, largely working within the bop tradition and bringing his own individualistic refinements to it; Hancock (and the rest of Davis’s quintet) combined inside and outside playing, commenting upon the bop tradition.38 These remarks, however, apply to the quintet’s performance of standards, where the chord changes are functionally tonal. The originals this band performed (many of them composed by saxophonist Wayne Shorter) frequently make use of modal, or “non-functional,” harmony, in which case we are far less likely to hear salient lines comprised of chordal extensions in terms of their underlying voice-leading motions. Many of the other points presented in Table 13-1 also frequently obtain in these circumstances, most notably point 11, a decreased emphasis upon more basic chord tones, which again can result in extended passages where roots

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are only implied. Furthermore, the modus operandi for these original tunes was often a very free approach that is referred to as “time, no changes.” As in classical repertoire, there is a rich spectrum of music that lies between common-practice jazz and fully atonal music. In the case of Evans (and others), Schenkerian analysis may be used to trace his movement along this spectrum. However, in order to adequately address music that is not entirely common-practice, certain principles of the Schenkerian approach must be relaxed.

Notes 1 For example, see Thomas Owens’s Bebop: The Music and Its Players, 3–4; David Baker’s How to Play Bebop, Volume 1: The Bebop Scales and Other Scales in Common Use, Preface (pages unnumbered); Scott DeVeaux and Gary Giddins’ Jazz, 607; and Richard Hermann’s “Charlie Parker’s Solo to ‘Ornithology’: Facets of Counterpoint, Analysis, and Pedagogy,” 222 (Hermann quotes Owens). In “Parallel Developments: Coltrane and Late-Romantic Music,” a paper given at the 2008 West Coast Conference on Music Theory and Analysis, I examine relationships between the development of the jazz and classical traditions using neo-Riemannian theory. 2 Of course, bebop was both a revolution and an evolution, and there were key musicians in the swing era that laid the groundwork for bebop. 3 Larson, “Schenkerian Analysis of Modern Jazz: Questions about Method,” 218. Larson’s hyphenation and the context would seem to indicate that by “post-modern jazz” he refers more to what is “post-bebop” than to what might be thought of as “postmodern” per se, such as “free jazz.” Other writers refer to “contemporary jazz” (see Baker’s Modern Concepts in Jazz Improvisation: A New Approach to Fourths, Pentatonics, and Bitonals), or simply reference the time period beginning around 1960 (see Keith Waters and Kent Williams’s “Modeling Diatonic, Acoustic, Hexa-tonic, and Octatonic Harmonies and Progressions in Two- and Three-Dimensional Pitch Spaces; or Jazz Harmony after 1960,” and Jack Chambers’s Milestones 2: The Music and Times of Miles Davis Since 1960; in the former, the text was written by Waters, and Williams created the examples). 4 For more on Schenkerian analysis of standards, see Forte’s The American Popular Ballad of the Golden Era: 1924–1950. Also see Larson’s review of Forte’s work in Music Theory Spectrum. 5 Larson, “Swing and Motive in Three Performances by Oscar Peterson,” 286. When dealing with multi-chorus solos, Larson illustrates ways in which performers delay or avoid tonal closure at the end of individual choruses, waiting until the conclusion of the improvisation to provide clear resolution. See, for example, Larson, Analyzing Jazz: A Schenkerian Approach, 66. Henry Martin is less inclined to perform voice-leading analyses of multi-chorus improvisations. When he discusses such solos in Charlie Parker and Thematic Improvisation, he argues that the background should usually correspond to only one chorus of a solo (in

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contrast to Larson), but also allows for some exceptions to this guideline; see pages 30–32. Also see Martin, “More Than Just Guide Tones: Steve Larson’s Analyzing Jazz—A Schenkerian Approach,” 130–31; idem, “Schenker and the Tonal Jazz Repertory,” 18; and Hermann, “Charlie Parker’s Solo,” 230–32. 6 Dizzy Gillespie reputedly was the first to use the word “vehicle” in this context. See Jerry Coker’s How to Listen to Jazz, 9. 7 Compositions following in this tradition are referred to as “contrafacts.” 8 Larson’s “The Art of Charlie Parker’s Rhetoric” is an exception to this (Martin’s “thematic improvisation” does not refer to the practice of directly referencing the original theme). 9 In one of his last publications, Larson acknowledges that unity does not stem only from thematic reference: “But I will grant that such global logic does not seem to require reference to the original melody or guarantee the sophisticated structures (e.g., hidden repetitions) admired in the common-practice works that theorists typically analyze.” It might be observed that the repeating harmonic progression of jazz is enough to produce a basic sort of unity, to say nothing of motivic coherence, whether the motives are derived from the theme (which they often are), or not. Larson, “Composition versus Improvisation?,” 259. 10 See Music Theory Online, volume 18, number 3 (September 2012), which is a festschrift for Larson, based upon a memorial conference held in the spring of 2012 at the University of Oregon. The 2013 conference of the Music Theory Society of the Mid-Atlantic featured a special session in memory of Strunk. Larson’s dissertation is entitled “Schenkerian Analysis of Modern Jazz.” Short-title references in this article to “Schenkerian Analysis” cite his article, not the dissertation. 11 Whereas “repertoire” typically refers to compositions, here it designates not just (or perhaps not at all) jazz compositions themselves, but rather to recorded performances of them. A standard tune in jazz—even if it may be successfully analyzed with a strict Schenkerian approach—may be performed in a limitless number of ways; some avant-garde artists choose to perform standards in a manner that at first appears to have little or no relationship to the original composition. 12 See Pellegrin, “On Jazz Analysis: Schenker, Salzer, and Salience,” chapters 1 and 2. In “Schenker and the Tonal Jazz Repertory,” Martin addresses the issue of “problematic compositions” in tonal jazz, proposing a number of additional Ursatz prototypes. He suggests that these models may be applied systematically to later jazz repertoire as well. (In these respects his work is reminiscent of Charles Smith’s contribution, “Musical Form and Fundamental Structure: An Investigation of Schenker’s Formenlehre”; see Pellegrin, “On Jazz Analysis,” pages 38–66, for discussion of Smith’s work). In addition, Martin discusses degrees of thematic reference in improvisation, noting that “in freer playing, one might have to begin with a deeper structural level of the original song to derive the solo” (his earlier work on Parker also deals with this issue). Ibid., 11. 13 Lerdahl, Tonal Pitch Space, 315.

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14 Larson’s approach to these tension tones relies heavily upon work of Strunk; see “Bebop Melodic Lines: Tonal Characteristics,” “The Harmony of Early Bop: A Layered Approach,” and “Linear Intervallic Patterns in Jazz Repertory.” 15 Larson, “Schenkerian Analysis,” 213; a different wording of this argument appears on page 214. 16 Larson, “Schenkerian Analysis,” 217. 17 For an analysis of a complete performance of Thelonious Monk’s “Green Chimneys,” see Pellegrin, “On Jazz Analysis,” pages 145–76. 18 For an introduction to this subject, see Keith Waters’s “Modes, Scales, Functional Harmony, and Nonfunctional Harmony in the Compositions of Herbie Hancock.” 19 An extreme example of this would be the practice known as “time, no changes,” sometimes used by Miles Davis’s quintet of the mid-1960s. See Waters’s The Studio Recordings of the Miles Davis Quintet, 1965–1968, and Todd Coolman’s dissertation, “The Miles Davis Quintet of the mid-1960s: Synthesis of Improvisational and Compositional Elements.” 20 For more on the way that thematic reference may be used in post-bop, see David Morgan’s “Superimposition in the Improvisations of Herbie Hancock.” 21 For an introduction to metrical issues in jazz, see Waters’s “Blurring the Barline: Metric Displacement in the Piano Solos of Herbie Hancock,” and Larson’s “Rhythmic Displacement in the Music of Bill Evans.” 22 This performance was originally released on Bill Evans Trio: At Shelly’s ManneHole, Hollywood, California. “Stella by Starlight” is a thirty-two measure composition in Bb major, with normative phraseology. For more on “Stella,” see the ninth volume of the Annual Review of Jazz Studies. The first half of this double-issue consists of the revised proceedings of an analysis symposium centered upon “Stella,” and a response by Forte. Larson investigates Evans’s threetrack recording of the tune on Conversations with Myself in one contribution (Larson also discusses this performance in “Composition versus Improvisation?”). 23 This version of the melody is taken from The Real Book. For a complete transcription of this performance, including bass and drum parts, see The Bill Evans Trio: Volume 2, 1962–1965, 105–22. Figure 1 is based upon this transcription, to which some minor corrections have been made. 24 The rhythm of the melody is taken from The Real Book, while the chord changes reflect what Evans actually plays in this performance of the head. The chords he plays, particularly in mm. 12–14, are substantially different than those found in The Real Book. 25 In m. 158, duration, metric placement, articulation, and resolution weaken the structural significance of the stable tone A natural in Evans’s melodic line, which is the leading tone. Rather, the A natural is a neighbor—an échappée in this case— to Ab (the raised ninth), which is a neighbor to Gb (the flatted ninth), and thus does not appear in Figure 13-3. This contrapuntal motion from the raised ninth to the flatted ninth, and finally down to the fifth of the tonic chord, is strongly directional, and is a staple of modern jazz. (Strunk and Larson deal with the raisedninth/flatted-ninth resolution as a special case. For more, see Strunk, “Bebop

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Melodic Lines,” 30; and Larson, “Schenkerian Analysis,” 215–16.) A similar situation may be found in a more simple context, when  occurs as an échappée in a melody line -- over V-I, although this is something of a special case as well. 26 The Bb occurring on beat 4 of measure 152 may be heard as thematic reference, but the connection between the C from measure 151 and the Db of measure 153 is stronger. Some other thematic tones in measures 152–60, while present in Evans’s right hand line, are similarly overshadowed by the middleground line given in Figure 13-3. That is why diagonal lines are not provided during these measures in Figure 13-1. 27 Curiously, after arriving at the final thematic tone F, Evans closes off the pitch space of this first chorus with a descent to 1. Lower neighbors embellish the onbeat -- in measure 159, with a brief move to a lower voice before the on-beat 
in measure 160. As shown in Figure 13-1, Bb is, in fact, also the pickup melody tone for the next chorus, with A succeeding it on the downbeat of the first measure of the form. Evans does begin his next phrase with this A, although it is an octave higher; the Bb seems to relate more to what precedes it rather than what follows it. 28 See Strunk, “Bebop Melodic Lines,” 112; and Larson, “Schenkerian Analysis,” 217. 29 Larson, “Schenkerian Analysis,” 216–17. 30 The numbers appearing in measure 24, particularly the 3 and the 13, may at first seem to be anomalous in terms of the pattern used for the rest of the example. However, this apparent aberration disappears if one remembers that the raised ninths appearing in the diagram are enharmonically equivalent to flatted thirds, and that the raised fifths are enharmonically equivalent to flatted thirteenths. Where there is a choice, the specific chord/scale tones given in this diagram attempt to reflect what Evans actually plays. For example, although the flatted ninth is also often played on a half-diminished ii chord, the natural ninth is given, as that is what Evans plays in these measures. In measure 31, Evans plays both the natural eleventh and the raised eleventh on the B major chord, but the Enatural is a chromatic lower neighbor (see note 27). In two cases, the diagram does not reflect what Evans plays. The major thirteenth (Db) he plays over the halfdiminished chord in measure 25 is not given, as the minor thirteenth is usually played; Evans actually plays both, but strongly emphasizes the Db. The salient flatted fifth (Eb) of measure 26 is not given; fifths—flatted-fifths on dominant chords—have been omitted from even-numbered measures for the sake of simplicity. (Odd-numbered measures of this strand would contain root doublings.) 31 The question of how the lack of a perfect fifth might affect the Strunk/Larson model is a separate issue, and one that applies to the half-diminished chord-types in use here as well. For an introduction to melodic-minor harmony, see Mark Levine’s The Jazz Theory Book, 55–77. For recent scholarship on the subject, see Dmitri Tymoczko, A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice; Waters, “Diatonic, Acoustic, Hexatonic”; and Waters, “Modes, Scales.”

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The major thirteenth of measure 153 (Db), essentially anticipates the resolution of the seventh, D, down to C# in the next measure. The Db thus would not resolve down by step, but rather would “resolve” by common-tone. Note also that the raised-ninth of measure 158 (Ab) must pass through the flatted ninth, as indeed it does, in order to resolve down by step (which is why, as noted above, Strunk and Larson deal with it as a special case). 33 This situation is somewhat similar to those encompassed by Schenker’s reachingover. Here, however, the tones of the ascending line are not stable chord tones, and the potential inner voice tones are often not found in Evans’s foreground lines. 34 In the first half of the excerpt, Evans comps in a manner closer to that of traditional bop style, where the left hand fills gaps in the right hand melodic line and offers other unobtrusive support. 35 In his biography, Bill Evans: How My Heart Sings, Peter Pettinger recalls Evans’s playing at his final performances at the Village Vanguard: “Now the notes were pouring out in a desperation born of inner fervor. Opening rhapsodies were massed in tiers, shifting tides around formal skeletons; piano solo codas were multilayered, like a one-man execution of Conversations with Myself.” In Bill Evans: Everything Happens to Me—A Musical Biography, Keith Shadwick refers to Evans’s “Brahmsian clouds of romanticism” in describing a recording drawn from these same performances at the Vanguard. Shadwick also quotes Harold Danko’s liner notes to the boxed set of these recordings as stating, “Rachmaninov comes to mind on these recordings even more than do Chopin, Ravel, and Debussy.” Pettinger, Bill Evans, 278; Shadwick, Bill Evans, 193, 195. 36 Morgan, “Superimposition,” 86. By “superimposition,” Morgan refers to “the technique by which an improviser plays a melody implying a chord, chord progression, or tonal center other than that being stated by the rhythm section.” Ibid., 69. 37 Ibid., 83. 38 See ibid., 86–88, for remarks concerning the significance of this commentary on the bop tradition.

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Works Cited Baker, David. How to Play Bebop, Volume 1: The Bebop Scales and Other Scales in Common Use. Van Nuys, California: Alfred, 2006. Originally published in Bloomington, Indiana: Frangipani Press, 1985. —. Modern Concepts in Jazz Improvisation: A New Approach to Fourths, Pentatonics, and Bitonals. Van Nuys, California: Alfred, 1990. The Bill Evans Trio: Volume 2, 1962–1965. Milwaukee: Hal Leonard, [2003?]. Chambers, Jack. Milestones 2: The Music and Times of Miles Davis since 1960. Toronto: University of Toronto Press, 1985. Coker, Jerry. How to Listen to Jazz. New Albany, Indiana: Jamey Aebersold, 1990. Originally published by Prentice-Hall in 1978. Coolman, Todd. “The Miles Davis Quintet of the mid-1960s: Synthesis of Improvisational and Compositional Elements.” Ph.D. diss., New York University, 1997. DeVeaux, Scott, and Gary Giddins. Jazz. New York: Norton, 2009. Forte, Allen. The American Popular Ballad of the Golden Era: 1924– 1950. Princeton, New Jersey: Princeton University Press, 1995. Givan, Benjamin. Review of Analyzing Jazz: A Schenkerian Approach, by Steve Larson. Journal of Music Theory 55, no. 1 (Spring 2011): 155– 60. Hermann, Richard. “Charlie Parker’s Solo to ‘Ornithology’: Facets of Counterpoint, Analysis, and Pedagogy.” Perspectives of New Music 42, no. 2 (Summer 2004): 222–63. Larson, Steve. Analyzing Jazz: A Schenkerian Approach. Harmonologia: Studies in Music Theory 15. Hillsdale, New York: Pendragon, 2009. —. “The Art of Charlie Parker’s Rhetoric.” Annual Review of Jazz Studies 8 (1996): 141–166. —. “Composition versus Improvisation?” Journal of Music Theory 49, no. 2 (Fall 2005): 241–275. —. Review of The American Popular Ballad of the Golden Era, 1924– 1950 by Allen Forte, The Music of Gershwin by Steven E. Gilbert, and Charlie Parker and Thematic Improvisation by Henry Martin. Music Theory Spectrum 21, no. 1 (Spring 1999): 110–21. —. “Rhythmic Displacement in the Music of Bill Evans.” In Structure and Meaning in Tonal Music: Festschrift in Honor of Carl Schachter, edited by Poundie Burstein and David Gagné, 103–22. Hillsdale, New York: Pendragon, 2006. —.“Schenkerian Analysis of Modern Jazz.” PhD diss., University of Michigan, 1987. —. “Schenkerian Analysis of Modern Jazz: Questions about Method.” Music Theory Spectrum 20, no. 2 (Autumn 1998): 209–241.

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—. “Swing and Motive in Three Performances by Oscar Peterson.” Journal of Music Theory 43, no. 2 (Autumn 1999): 283–314. Lerdahl, Fred. Tonal Pitch Space. New York: Oxford University Press, 2001. Lerdahl, Fred, and Ray Jackendoff. A Generative Theory of Tonal Music. Cambridge, Massachusetts: The MIT Press, 1983. Levine, Mark. The Jazz Theory Book. Petaluma, California: Sher Music, 1995. Love, Stefan C. “Subliminal Dissonance or ‘Consonance’? Two Views of Jazz Meter.” Music Theory Spectrum 35, no. 1 (Spring 2013): 48–61. Martin, Henry. Charlie Parker and Thematic Improvisation. Studies in Jazz 24. Institute of Jazz Studies, Rutgers—The State University of New Jersey / Lanham, Maryland: Scarecrow Press, 1996. —. “More Than Just Guide Tones: Steve Larson’s Analyzing Jazz—A Schenkerian Approach.” Journal of Jazz Studies 7, no. 1 (Spring 2011): 121–44. Martin, Henry. “Schenker and the Tonal Jazz Repertory.” Dutch Journal of Music Theory 16, no. 1 (2011): 1–20. Morgan, David. “Superimposition in the Improvisations of Herbie Hancock.” Annual Review of Jazz Studies 11 (2000): 69–90. Owens, Thomas. Bebop: The Music and Its Players. New York: Oxford University Press, 1995. Pellegrin, Rich. “On Jazz Analysis: Schenker, Salzer, and Salience.” PhD diss., University of Washington, 2013. Pettinger, Peter. Bill Evans: How My Heart Sings. New Haven, CT: Yale University Press, 1998. The Real Book. Sixth ed. Milwaukee: Hal Leonard, [2004?]. Shadwick, Keith. Bill Evans: Everything Happens to Me—A Musical Biography. San Francisco: Backbeat Books, 2002. Smith, Charles. “Musical Form and Fundamental Structure: An Investigation of Schenker’s Formenlehre.” Music Analysis 15, no. 2/3 (July–October 1996): 191–297. Strunk, Steven. “Bebop Melodic Lines: Tonal Characteristics.” Annual Review of Jazz Studies 3 (1985): 97–120. —. “The Harmony of Early Bop: A Layered Approach.” Journal of Jazz Studies 6 (1979): 4–53. —. “Linear Intervallic Patterns in Jazz Repertory.” Annual Review of Jazz Studies 8 (1996): 63–115. Tymoczko, Dmitri. A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice. New York: Oxford University Press, 2011.

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Waters, Keith J. “Blurring the Barline: Metric Displacement in the Piano Solos of Herbie Hancock.” Annual Review of Jazz Studies 8 (1996): 19–37. —. “Modes, Scales, Functional Harmony, and Nonfunctional Harmony in the Compositions of Herbie Hancock.” Journal of Music Theory 49, no. 2 (Fall 2005): 333–357. —. The Studio Recordings of the Miles Davis Quintet, 1965–1968. New York: Oxford University Press, 2011. Waters, Keith J., and J. Kent Williams. “Modeling Diatonic, Acoustic, Hexatonic, and Octatonic Harmonies and Progressions in Two- and Three-Dimensional Pitch Spaces; or Jazz Harmony after 1960.” Music Theory Online 16, no. 3 (August 2010).

Discography Catalog numbers reference the most recent release of material. The dates given are the dates of recording. Evans, Bill. Bill Evans Trio at Shelly’s Manne-Hole. Riverside OJCCD 263-2. Hollywood: May 14 and 19, 1963. —. Conversations with Myself. Verve 314 521 409-2. New York: February 6, 9, and May 20, 1963.

CHAPTER FOURTEEN A CRITICAL COMPARISON OF ARISTOXENUS’ AND PTOLEMY’S GENERA MATTHEW E. FERRANDINO One of the earliest and clearest examples of the struggle between theoretical and practical concerns in music theory discourse can be seen through a comparison of the harmonic treatises of Aristoxenus’ Elements of Harmony (fourth century B.C.E.) and Ptolemy’s Harmonics (second century C.E.).1 Both theorists define methods of tuning various tetrachords, or genera, as a basis for melodic composition. However, their methods of creating these tetrachords differ drastically, and I argue that the Aristoxenian method represents a practical approach to musical scholarship while Ptolemy’s method represents the theoretical. By looking first at the formation of genera defined by each theorist, I argue that Aristoxenus is representative of practical music theory while Ptolemy is archetypical of the theoretical side of the discipline. Aristoxenus’ method of constructing tetrachords is presented through abstract relations of a tone, purposely bereft of discrete mathematics and is constantly referred to the singing of melodies. Ptolemy’s method is an expansion of the Pythagorean approach; using integer ratios to define the intervallic distance between pitches. Rather than tracing a critical history through documents I focus on the original manuscripts in order to present a more nuanced comparison—focusing on each theorist’s methodology and presentation of tuning genera. A comparison of the resultant tetrachords and their variations will show that the auditory discrepancy between those constructed by Aristoxenus and Ptolemy is minimal to non-existent, therefore the fundamental differences between the two theorists are based on their intent: practical versus theoretical.

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Aristoxenus’ Formation of the Genera Aristoxenus utilizes two Pythagorean ratios to define the intervals used in his genera: 4:3, the perfect fourth, and 9:8, the tone—which is the difference2 between the intervals of the fifth and the fourth.3 The interval of a fourth remains a constant outer interval throughout the genera and their variations. The tetrachord is introduced by Aristoxenus using Greek pitch names, Hypate, Parahypate, Lichanos, and the Mese. 4 The outer pitches, the Hypate and the Mese, are at a fixed Pythagorean interval of a fourth, while the two inner pitches are movable. Aristoxenus acknowledges the infinite possible positions of the two inner pitches but limits their positions from a practical point of view: “The voice can not distinctly produce an interval even smaller than the smallest diesis nor can the hearing detect one in such a way to grasp what part is either of a diesis or of any of the other intervals which are known.”5 The smallest diesis that is mentioned refers to the enharmonic diesis, or 1/4 of a tone—a tone being 9:8.6 Without going into too much detail about the divisions of a tone at the moment, we should first understand the different types of genera to be dealt with. Aristoxenus writes, “There are three genera of melodies, the diatonic, the chromatic, and the enharmonic. This we may lay down, that every melody must be diatonic, chromatic, or enharmonic, or blended of these kinds, or composed of what they have in common.”7 Within each of the three basic types Aristoxenus defines variations, or shades, of each with the exception of the enharmonic. The chromatic genus can be shaded three different ways: soft, hemiolic, and tonic, and the diatonic genus two different ways: soft and tense—giving us a total of 6 possible distinct genera depending on the pitches of the Parahypate and the Lichanos. In order to reconstruct each of the genera we must first look at how Aristoxenus deals with divisions of the tone. He explains that: A tone is the excess of a fifth over the fourth; the fourth consists of two tones and a half. The following fractions of a tone occur in melody: the half, called a semitone; the third, called the smallest chromatic diesis; the quarter, called the smallest enharmonic diesis. No smaller interval than the last occurs in melody.8

While the aforementioned fractions are the divisions of the tone employed, we must also be prepared to deal with an eighth of a tone and a sixth of a tone. These divisions of the tone do not occur as 1/6 tone or 1/8 tone, but as multiples of those divisions. Before presenting an intervallic account of each genus, Aristoxenus first provides practical ranges for the intervals

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between the Hypate and Parahypate, and the Lichanos and the Mese: the Parahypate may be placed between the intervals of 1/4 of a tone up to 1/2 of a tone above the Hypate. Similarly, the Lichanos may be placed between the interval of a tone and a ditone (2 tones) below the Mese9. Example 14-1 illustrates these definitions as well as the actual intervals in each of the six genera,10 numerical values are given as multiples of a tone.

Example 14-1, Interval ranges and interval distribution in Aristoxenus’ genera

In order to more fully appreciate the subtleties between the six genera we can express them in terms of a more familiar measurement of intervallic distance: cents.11 This will allow us to not only compare Aristoxenus’ genera to those constructed by Ptolemy, but also to give a point of reference to our modern conventional equal-tempered system. Example 14-2 outlines the necessary calculations for this process.

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Example 14-2, Aristoxenus’ intervals defined in cents

We can now redefine the six variations of genera, using cents and deviations from the more familiar 12-tet, shown in Example 14-3.12 An important discretion arises here between the interval sum of two and a half tones, 510 cents in Example 14-3, and the interval of a fourth (4:3) defined in Example 14-2 as 498 cents. This issue will be discussed later on with Ptolemy’s genera, yet it is interesting to note that two disjunctly paired genera (separated by a tone, 204 cents) yields an excess of the octave (2:1, or 1200 cents). The discrepancy: 510 + 204 + 510 = 1224, and creates an excess over the octave of 24 cents—or the didactic comma. The same amount of discrepancy occurs from tuning a 12-note chromatic scale based on fifths (3:2), (Pythagorean tuning). Therefore, it is important to keep in mind that Aristoxenus’ method of tuning is essentially still a Pythagorean system, but presented as divisions of the tone rather than ratios. However, Aristoxenus does not acknowledge the mathematical discrepancies that arise from his method, but rather insists that his description is grounded in musical practice.

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Example 14-3, Aristoxenus’ genera using cents

Aristoxenus’ initial dissatisfaction with the mathematical approach presented by the Pythagoreans is summed up succinctly by Andrew Barker, “Their [the Pythagoreans’] mistake was to look in the wrong place for the principles on which their arguments depended, drawing them from mathematics and theoretical physics rather than seeking principles ‘perceptually evident to those experienced in music.’ As a consequence, their arguments are irrelevant.”13 The arguments presented by Aristoxenus empirically order musical accuracy over mathematical accuracy. Ptolemy,

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as we shall see, attempts to present his argument keeping both musical and mathematical accuracies of equal importance.

Ptolemy’s Formation of the Genera We have seen that Aristoxenus’ method of defining the genera revolved around the tone (9:8) and various divisions of it. Ptolemy’s system is centered on ratios, more specifically, epimoric, or superparticular, ratios. Epimoric ratios take the form of: (x+1):x. Stated more simply, the two integers of the ratio have a difference of 1. This definition incorporates the intervals so far mentioned: 2:1, 3:2, 4:3, and 9:8. As we saw, there is a mathematical discrepancy in Aristoxenus’ method of equating the interval of a fourth (4:3) to that of two and a half tones (9:8). Ptolemy claims that, “[Aristoxenians] are mistaken, furthermore, about the measurement of the first and smallest concord, composing it as they do from two tones and a half.”14 Ptolemy goes on to explain that in order to complement the 4:3 interval, two tones must be added to an interval that is slightly less than half of a tone, which he defines as the Leimma. The interval of a Leimma can be explained through the method previously discussed. Recalling our comparison of Example 14-3 and Example 14-2, there existed a discrepancy of 12 cents between the calculated interval of a fourth (4:3) and the sum of two and a half tones. If we then correct this discrepancy by removing 12 cents from the interval of a semitone (102 cents) we can define the interval of the Leimma as 90 cents. An alternate approach would be to take the difference between the fourth and the sum of two tones: (4:3)/[(9:8)*(9:8)] = 256:243.Therefore the Leimma can be defined as the ratio of 256:243 or, again, 90 cents. Fundamentally, Ptolemy’s method of constructing genera is similar to Aristoxenus’ in that the interval of a fourth is ‘fixed’ between the outer pitches of the tetrachord. The variations between genera are similarly defined by the placement of the inner pitches, previously referred to as the Parahypate and the Lichanos. Using the epimoric ratio of 4:3 as the progenitor for his intervals, Ptolemy is able to define eight variations of genera: one enharmonic, two chromatic, and five diatonic. The concept behind his construction is based on the theory that an epimoric ratio can be expressed as the sum of two smaller epimoric ratios. Example 14-4 presents an objective algebraic interpretation of this theory.15

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Example 14-4, Epimoric ratio pair relations from progenitor 4:3

Nearly all of the ratios that Ptolemy employs can be defined this way from the progenitor ratio of 4:3. Given the assumption that the 4:3 epimoric ratio has three distinct epimoric ratio pairs (total of six ratios) whose sum equates back to 4:3, each of these ratios can themselves be expressed as a sum of two epimoric ratio pairs. Example 14-5 clarifies this point.

Example 14-5, Expressing an epimoric ratio as the sum of two smaller epimoric ratios

Using the same formula from Example 14-2, we can reinterpret these intervals in terms of cents. Example 14-6 shows the first ‘family’ of pairs from 4:3, and Example 14-7 shows the secondary ‘family’ of pairs that sum to the ratios of the first family.

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Example 14-6, Epimoric ratio pairs from 4:3, defined in cents

Example 14-7, Epimoric ratio pairs from pairs of 4:3, defined in cents

Ptolemy suggests eight variations on the three basic forms of the genera: one enharmonic; two chromatic: soft and tense; and five diatonic: soft, tonic, tense, even, and ditonic. All of the above mentioned variations utilize the epimoric ratios defined thus far with the exception of the diatonic even, and diatonic ditonic. The diatonic ditonic genus has been already suggested by the division of the tetrachord into two tones (9:8) and the Leimma (256:243), which is not an epimoric ratio. The diatonic even genus introduces a unique epimoric ratio, 11:10, which can be found by tripling the 4:3 ratio: 12:9, and then finding the mean epimoric ratio by subtracting 1 from the higher number and adding 1 to the lower number resulting in the ratio 11:10. Therefore the diatonic even genus consists of subsequent ratios 10:9, 11:10, and 12:11. Example 14-8 represents Ptolemy’s eight genera16 with cents and deviation from familiar 12tet pitches.

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Example 14-8, Ptolemy’s genera using cents

We can now look at the dissatisfaction Ptolemy has with the previous Pythagorean tradition and Aristoxenian tradition.17 Ptolemy’s general problem with the Pythagorean tradition is their definition of concordance, or consonance. To put simply, he believes that the octave and a fourth interval, (8:3), is indeed a consonant interval, despite the Pythagorean claim that it is dissonant due to the numerical values of the ratio. “In spite

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of what the Pythagoreans hypothesize about the consonances, the diapason plus the diatesseron [octave plus a fourth] is clearly a consonance in every way and brings shame upon the ratio they applied it to.”18 Without going too far into Ptolemy’s particular nomenclature for redefining consonance, he qualifies them as belonging to one of three categories, the highest including the interval of an octave, the second highest that of the fifth and the fourth, and lastly everything else that is the tone and its divisions (he does not use the term dissonant for this last category). The Aristoxenians are chided not for their intentions of seeking a more musically accurate system, but for their inability to then explain their system rationally with mathematics. “If hearing followed the lead of reason to the greatest degree possible, while on the other hand the outstanding experts in the subject condemned it, though they are unable, by themselves, to initiate an investigation of the rational divisions, and neither do they think fit to try and discover those that are displayed by perception.”19 The system of genera Ptolemy has constructed is arguably similar in its result to that of Aristoxenus’ system, but his method is justified in a “rational” mathematical foundation.

Brief Comparison of the Genera If we compare the diatonic soft genera as defined by Ptolemy and Aristoxenus, there is a small deviation in resultant pitches. Given that the starting pitch that each tetrachord is built on is the same, and that the outer interval of a fourth has a discrepancy of 12 cents, the second note of the diatonic soft genera has a discrepancy of 17.5 cents, and the third a discrepancy of 11.9 cents. These discrepancies would be considered insignificant from Aristoxenus’ claim that “the ear is unable to discern an interval smaller than 1/4-tone.” Comparing other genera similarly named, the discrepancies remain indiscernible with the exception of the chromatic soft genera. In this case the difference in the third pitch of 46.4 cents is close to 1/4-tone. Rather than interpreting this as a fault of either Ptolemy’s or Aristoxenus’ method of tuning, it is more likely that the two genera, though similarly named, do not correspond to the same genus. Since the interval of difference is close to that of a 1/4-tone difference from Aristoxenus’ system, it is possible that this result in Ptolemy’s system is an adjustment made through the change of practice. It is beyond the scope of this paper to explore a reinterpretation of Ptolemy’s genera in terms of Aristoxenian methods, but it seems likely that the resultant system using divisions of the tone would be audibly similar to one using ratios. Example 14-9 gives a comparison of the four similarly named tetrachords.

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Example 14-9, Comparison, in cents, of Aristoxenus’ and Ptolemy’s genera

The P(t)olemic The genera defined by Ptolemy leave little room for interpretation. Each interval is explicitly defined in terms of epimoric ratios, with the exception of the Leimma. Further analysis can prove that the 4:3 ratio itself is an epimoric subset of the octave, 2:1—with its compliment being the perfect fifth, 3:2. That is to say that 4:3 multiplied by 3:2 gives us the ratio of 12:6; which can then be reduced to 2:1. We can argue then that the intervallic content of seven out of the eight genera defined by Ptolemy are variations on the purest consonance: the octave. Furthermore, Ptolemy argues that his genera are practical for categorizing music of his time. Given the limited resource of musical examples from this time, it is impossible to argue the validity of his claim. Despite his intention of combining the mathematical significance of the Pythagorean tradition with the practical application of the Aristoxenian tradition, his highly mathematical ratios and definitions place him clearly in the theoretical, or Pythagorean, approach to defining music. That is not to say there is not a practical application of Ptolemy’s genera. For example, Gioseffo Zarlino favored the diatonic tense tuning for its close to pure thirds: “Zarlino advocated a scale that favored the thirds and fifths without disturbing the true intonation of most of the perfect consonances. Ptolemy called it

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[Diatonic Tense], because the third of the four notes of the tetrachord was sharper than in other versions.”20 Aristoxenus, on the other hand, is ambiguous enough in his methodology to leave room for interpretation and speculation on his formation of genera. In the beginning of his treatise he purposely absolves the connection with the Pythagorean tradition of numerical significance and ratios in application to music. His definition of the tone being defined abstractly as the difference of the fourth and the fifth, without explicitly indicating 4:3 and 3:2 ratios for said intervals, is certainly open to interpretation in terms of its application. From his initial resolve to dismiss mathematical definitions, one can speculate that his intentions are of a more practical nature. Continuing on this speculation, it can be argued that the tone itself, as defined by Aristoxenus, does not maintain a linearintervallic value, as suggested by Examples 14-2 and 14-3, but may in fact be variable in size depending on how and where it is applied in the tetrachord. “Aristoxenus knew very well that the quality of sound had to be distributed in equal parts, not the quantity of the line, string, or space. He was operating as a musician on a sonorous body, not as a pure mathematician on a continuous quantity.”21 Several other 16th Century theorists—including Giovanni Maria Artusi—echo Galilei’s opinion of Aristoxenus as well.22 The same allowance of variation can be applied for Aristoxenus’ divisions of the tone and their use in specific genera. We can posit that Aristoxenus was aware of the mathematical discrepancy between the 4:3 ratio of the fourth and the sum of two and a half tones, and that a practical method of pitch production is implied, which compensated for the difference based on the skills acquired through practicing music. Barker writes that “[Aristoxenus] insists that students of the subject must become familiar with melodies of many sorts, and must train their hearing, their memory, and their interpretative sensitivity to grasp what is going on in any melody as it unfolds.”23 The difference between the two intervals mentioned, 12 cents, is below the smallest discernible interval suggested by Aristoxenus, that of the enharmonic diesis: 1/4-tone or 51 cents. This discrepancy was not accounted for in Aristoxenian practice, due to the inability to explain the compensation mathematically, and remains an ambiguous aspect of his treatise, one which Ptolemy primarily finds fault in. Both methodologies of tuning the genera are presented as practical interpretations of music, but it is in their conception and application where the dichotomy of practical versus theoretical becomes an issue. It is clear that Aristoxenus’ intent is to provide a method of tuning and composition

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that is graspable and accessible to musicians, more specifically singers. The division of the tone into 1/2, 1/3, and 1/4 is a much simpler concept to apply in the singing of music, rather than attempting to conceive of the distinction between 21:20 and 22:21. On the other hand, Ptolemy succeeds in his intent to satisfy both musical accuracy as well as a rational mathematical accuracy in his method of tuning. However, Ptolemy’s system is one that is much less accessible to the practicing musician, specifically the singer. The epimoric ratios stem from a mathematical foundation, and though arguably possible, the application of the ratios in practice is absurd. The underlying distinction between Aristoxenus’ treatise and Ptolemy’s treatise is therefore best understood from the author’s intent: Aristoxenus writing a practical manual of music, and Ptolemy writing a mathematical account of what music is. Both are based on the practice of their respective times, but the dichotomy of their intentions is practical versus theoretical.

Notes 1

The polemic is exemplified in the 16th century with Gaffurio’s critique of Aristoxenus, in favor of Ptolemy, in De harmonia musicorum instrumentorum opus, 1518, and Vincenzo Galilei’s defense of Aristoxenus in his Dialogo, 1581. 2 It is important to note that “difference” when discussing ratios refers to division, not subtraction. 9:8 is therefore the result of (3:2)/(4:3). Similarly, a sum of two ratios would imply a multiplication not an addition. 3 Aristoxenus refrains from defining intervals as Pythagorean intervals, but in order to create a meaningful mathematical comparison between his system and Ptolemy’s, it is necessary to assume that these ratios are implied. 4 Here I have reordered the pitch names from low to high, the Mese being a fourth above the Hypate. In Aristoxenus’ original treatise they are presented from high to low. 5 Andrew Barker (ed.), Greek Musical Writings, Volume II: Harmonic and Acoustic Theory (Cambridge University Press, 1989), 135. 6 Aristoxenus also describes a chromatic diesis, 1/3 of a tone. For the purpose of this paper the term “diesis” will not be used, as Aristoxenus occasionally omits qualifying the term as enharmonic or chromatic in his treatise. I will present them in terms of their relation to the tone, 1/4 tone or 1/3 tone. 7 Aristoxenus, Elementa Harmonica, translated and edited by Henry S. Macran (Oxford: Clarendon Press, 1902), 45. 8 Aristoxenus, 199. 9 Aristoxenus, 181. 10 I have omitted the step-by-step creation of each genus since the nomenclature is beyond the scope of this paper. For a more in depth construction see: Barker, Greek Musical Writings, 164-165.

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The term ‘cents’ is accredited to English mathematician Alexander John Ellis, who first described the calculation of cents in Journal of the Society of Arts 28 (March 5, 1880): 295. 12 Perhaps the most well-known encyclopedia of historical tuning systems is James Murray Barbour’s 1951 dissertation “Tuning and Temperament: A Historical Survey.” The calculations provided in Example 14-3, however, do not match Barbour’s. For example, his calculation for Aristoxenus’ chromatic tonic genus gives intervals (in cents) of [0; 89; 182; 498], whereas Example 14-3 gives the intervals as [0; 51; 255; 510]. Barbour does not elaborate on Aristoxenus at all, but it is clear that his calculations for the outer fourth between the Hypate and the Mese is based on the 4:3 ratio of a fourth, 498 cents, rather than the sum of two and a half tones that Aristoxenus puts forth, which results in 510 cents. It is also interesting that the two inner intervals that he calculates are not explainable through the various divisions of the tone in cents from Example 14-2. His citations include the Macran translation of Aristoxenus that has been cited in this paper as well, so the discrepancy is not one of translated editions. See James Murray Barbour, Tuning and Temperament: A Historical Survey (Michigan State College Press, 1951), 17. 13 Andrew Barker, The Science of Harmonics in Classical Greece (Cambridge University Press, 2007), 167. 14 Barker, Greek Musical Writings, 295. 15 A mathematical proof of this interpretation is not provided for this theory, so for the moment it remains a postulated algebraic representation of Ptolemy’s method. 16 For Ptolemy’s full outline for each genus see: Ptolemy, Harmonica, translated by Jon Solomon (Leiden: Koninklijke Brill NV, 2000), 47-59; and Barker, Greek Musical Writings, 306-314. 17 Since Ptolemy’s Harmonica is written almost 500 years after Aristoxenus, his reference to Artistoxenians is not so much directed at Aristoxenus himself, but those that inherited and elaborated upon his methodology in later years. 18 Ptolemy, 19-20. 19 Barker, Greek Musical Writings, 311. 20 See Claude Palisca’s introduction from: Vincenzo Galilei, Dialogue on Ancient and Modern Music, translated by Claude V. Palisca (Yale University Press, 2003), xxxv. 21 Galilei, 127. 22 For a full discourse on 16th Century theorists’ defense of Aristoxenus, see: Claude V. Palisca, “Aristoxenus Redeemed in the Renaissance,” Revista de Musicaologia 16/3 (1993): 1283-1293. 23 Barker, The Science of Harmonics in Classical Greece, 181.

CHAPTER FIFTEEN SPACIOUSNESS OR EVENNESS? A THEORY OF HARMONIC DENSITY IN ANALYZING 20THAND 21ST-CENTURY MUSIC YI-CHENG DANIEL WU1 To make 20th- and 21st-century harmonic vocabulary more accessible, composers and theorists have derived different approaches to define the meaning of a chord. These approaches can be categorized into two trends—measuring the chord’s similarity and defining its harmonic quality. In the first trend, theorists draw attention to the relationship between any two chords with respect to their set-class (sc) representatives and interval-class (ic) vectors, proposing various arithmetic formulas to measure the contents of the two ic vectors. The result is a decimal number reflecting the degree of similarity between those two ic vectors and their associated scs. These measurements include Robert Morris’s ASIM, Eric Isaacson’s IcVSIM, and Michael Buchler’s SATSIM, among others.2 The second is fairly aligned with our traditional understanding of a chord, for it carries on the notions of consonance and dissonance to define the quality of a chord. This trend includes the theories proposed by composers such as Paul Hindemith, Ernst Krenek, and George Perle, among others.3 In a similar fashion to the second trend but with a different and innovative approach, Joseph Straus extends his offset number derived from fuzzy transformational voice leading to examine harmonic quality.4 Unlike the previous definitions, Straus’s theory is geared towards another facet of a chord, which is the density.5 His premise is that the offset number will increase and decrease its value based on the density of a chord. He uses two categories of terms to define the harmonic density— “compactness, denseness, chromaticness” for chords with a small offset number projecting a saturated texture packed by compact ics, and “dispersion, spaciousness, evenness” for those with a large offset number projecting a hollow texture stacking up spacious ics. 6 All of these terms should

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consistently aid the listener in hearing a chord with respect to its overall level of density, such as high or low. But if we visualize the images that they project, the last one— the evenness— draws an essentially different picture from all the others. It reflects the proportion, not the level, of the density, which describes a chord with a balanced intervallic structure whose pcs are generated by the same ic. Figure 15-1 shows a problem with mixing level and proportion to interpret a chord’s density, as realized particularly by Straus’s terms of evenness and spaciousness.




Figure 15-1, from Straus, “Voice Leading in Set Class Space,” p. 69


Figure 15-1 reproduces Straus’s example, in which all twelve trichords are measured against sc 3-1, resulting in offset numbers 0–6. The smallest 0 and the largest 6 respectively represent sc 3-1 and sc 3-12. Comparing these two offset numbers, we can agree with Straus that the larger the offset number, the lower density the chord, because sc 3-12 is more spacious than sc 3-1. But it is illegitimate to characterize sc 3-12 with a larger offset number as being proportionally more even than sc 3-1, because the pcs in both scs are all generated by the same ics,7 which make these two scs both structurally balanced chords.
Thus, the offset number
can only reflect the level of the density, not the proportion, for the more

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spacious chord may not be necessarily more even, and the less spacious chord, contrarily, may not be truly less even. Using the problem addressed in Figure 15-1 as a point of departure, I aim to extend and clarify Straus’s definitions of harmonic density— particularly his evenness and spaciousness. I begin with a brief survey of Straus’s fuzzy transformational voice leading and offset number, and then proceed to redefine his concepts of spaciousness and evenness by establishing a clear boundary between them. From the ground up, I use my redefinitions to first examine the density of all ics, finding representatives that can more accurately reflect the sense of harmonic evenness and spaciousness, and then use these redefined ics as my references to develop a measurement examining a chord’s textural density articulated by its level and proportion. To demonstrate the practical advantage of this measurement, analyses of the works of Arnold Schoenberg, Alban Berg, Ruth Crawford, and György Kurtág will conclude this essay. Straus asserts that there exists a fuzzy transformational voice leading space between two different scs. He uses the term maximally uniform to indicate the quality of fuzzy transposition and maximally balanced to signify that of fuzzy inversion. The principle of Straus’s transformational voice leading is to transform the greatest number of pitches in one chord onto their correspondents in the next by the same semitonal distance, while the remaining pitches are related by a dissimilar one. The difference of these two distances results in an offset number (in mod 12, usually marked with parentheses). It is possible that there will be several fuzzy transformations at the same time all relating the greatest number of pitches in the adjacent chords, and we must determine which one is preferable to the others. Straus suggests comparing their offset numbers. He chooses the smallest one, which he calls the minimal offset number, to be his ideal candidate. For Straus, this number complements the most uniform or balanced quality and can create a sense of parsimony between adjacent chords. The smaller the offset number, the more parsimonious the voice leading will be between two chords.8 Based on the concept of offset number, Straus proposes a harmonic measurement to test the density of a chord— or, in his words, “the degree of chromaticness.” He begins by measuring all the dyads against sc 2-1 (see Figure 15-2).




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Figure 15-2, from Straus, “Voice Leading in Set Class Space,” p. 68

As dyads move away from sc 2-1, their offset numbers become larger.9 In this figure, the pcs are either extremely packed in the sc appearing on the top layer (i.e., sc 2-1), or dispersed in that on the bottom (i.e., sc 2-6). Straus uses these extremes to define sc 2-1 with the smallest offset number as projecting the highest level of density— “compactness, denseness, chromaticness,” and sc 2-6 with the largest as projecting the lowest level— “dispersion, spaciousness, evenness” (emphasis mine).10 Since the first three terms “compactness, denseness, chromaticness” convey the same image of a small harmonic space, I choose the one that occurs the most frequently in Straus’s article throughout my discussion— chromaticness. Likewise, I use his spaciousness to consistently describe large harmonic space and further compare it with evenness. One issue arises when we replace all the dyadic scs with their corresponding ics— ic1 formed by sc 2-1 represents Straus’s chromaticness, and ic6 formed by sc 2-6 represents what he asserts as both evenness and spaciousness. In fact, most of this statement is not particularly intriguing, for we commonly acknowledge that ic1 and ic6 respectively project the most packed and dispersed spaces. But within a limited environment framed by ic1 and ic6, why and how is ic6 more even than the other five ics1–5? To consolidate my point, Figure 15-3 unfolds all the spaces of ics1–6 and compares them.11

Spaciousness or Evenness? A Theory of Harmonic Density

Figure 15-3, comparing the spaces of ics1–6

293





We can assert that ic6 stretches out into the most expanded space, which is spacious enough to cover ics1–5. But our definition becomes illegitimate once we replace Straus’s word “spacious” with his “even”— “ic6 stretching out into the most expanded space, which is even enough to cover ics1–5.” The cause of this illegitimacy is that evenness does not reflect whatsoever the high or low level of harmonic density as spaciousness does, and explaining the role of evenness in the context presented in Figure 15-3 requires further information. Figure 15-4 distributes all the dyadic scs into an octave projected by a clock, a schema used by most theorists to represent pc space.










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Figure 15-4, the dyadic scs projected on an octave represented by a clock


Each clock contains two marked pcs dividing the circumference into two curves. All the curves in clocks (1) to (5) are uneven, and we choose the shorter ones to represent their corresponding ics. Only those in clock (6) are even, and either one forms the representative of ic6. In this context, with the reference of the octave, we can define ic6 is more even than the other ics1–5.12 Since this schema is a necessary ground to define the sense of evenness, the bottom of Figure 15-5 unfolds a clock into a horizontal line projecting an octave (sc1-1) and its ic representative of ic0, which serves as a reference to measure the evenness of the other ics. (Note that hereafter ic0 only ever refers to an octave, not a unison. This will be thoroughly explained once we arrive at Example 15-1.)





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Offset from 1-1[00]

chromatic

2-1 01

0

2

2-2 02

0

3

2-3 03

0

4

2-4 04

0

5

2-5 05

0

6

2-6 06

0

0

1-1 00

ic1

1

0

ic2

0

2

ex

1

pa 3

0

nd

ic3

ing the 0

sp

4

ic-

ic4

e

ac

even spacious

ic5

0

5

ic6 6 ic0/octave (referential ic0-line) 0

ic6

0 0

11

1 2

10 0

9

3 8

4 7

6

5

Figure 15-5, ic0 used as a reference to measure the evenness of the other ics


Each ic breaks down our referential ic0-line into two parts distinguished by different colors, black and gray. The former represents the shorter side of the curve derived from its corresponding clock in Figure 15-4 and the ic. The latter represents that of the longer side. Only sc 2-6 evenly dichotomizes our referential ic0-line, resulting in two black equal parts. I choose the one on the left to consistently represent ic6. Based on the length of the black parts, I can use Straus’s chromaticness and spaciousness to prescribe the spaces of ic1 and ic0, and evenness is assigned to ic6, for it proportionally bisects ic0. Attention must be paid to two resultant peculiarities in this figure. The first is the detachment of Straus’s evenness from his spaciousness. These two definitions no longer belong to the same category; they represent two different ics. Due to such detachment, we must read Figure 15-5 in two parallel ways simultaneously in terms of the harmonic density: (1) the level, and (2) the proportion. From the aspect of level, following the pale dotted arrows the ics gradually expand their spaces and decrease the levels of their densities. Proportionally, we measure from the interval that can evenly dichotomize
our referential ic0-line into two equally balanced structures— that is, ic6.
The second peculiarity is an issue raised by the offset number. Measuring all the scs in relation to sc 1-1,13 the outcome shows that sc 1-1 receives the smallest value 0, which suggests Straus’s most chromatic density. But this outcome drastically contradicts my earlier spatial prescription of the ic0, which appears to be the most spacious. In this case, the offset number cannot truly reflect the harmonic density of all seven ics0–6. This issue, hence, necessitates a theory that more definitively and




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accurately interprets the harmonic density of all the ics, and derives from it a method capable of measuring the density of a chord. *** The following section uses Figure 15-5 as my theoretical ground to define the density of all ics. Example 15-1 orders the seven ics from Figure 15-5 according to their densities from high to low.

chromatic even spacious



ics 1 2 3 4 5 6 0
octave


harmonic density high low
Example 15-1, ordering the seven ics according to their level and proportion of harmonic densities


Within this spatial environment, ic6 represents a proportionally even density. I must stress that in the framework of Example 15-1 we hear ic0 as always representing an octave, never a unison. Which means: the octave is not inversionally equivalent to a unison. With this particular condition in mind, and in the theoretical spectrum of this paper, we hear the unison as a “tone repetition,” which does not form an interval.14 One crucial benefit supports us in not recognizing ic0 as a unison. Conventionally, most theorists order the seven ics from 0 to 6 as shown in Example 15-2.

chromatic ?

ics harmonic density

0

1

high

Example 15-2, the conventional ic-ordering

2

3

4

5

6 low



They hear ic0 and ic6 as respectively projecting the highest and lowest density. But this creates a problem if we keep Straus’s sense of chromaticness to define ic0. Although it may represent two pcs that have

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no space in between and create a unison, we do not call the unison “chromatic.” Thus, grounded in the conditions that 1) the unison is not an interval but a tone repetition, and 2) ic0 only ever refers to an octave, we can use Straus’s chromaticness to define the density of ic1 as suggested in Example 15-1. To reinforce the concept of ic0 as always possessing the lowest density or the most spacious ic, I again incorporate a clock schema and slightly modify the traditional measurement of an ic in that schema. I set up two rules to measure an ic. First and foremost, no pc ever remains stationary. Second, we consider the shortest distance between two pcs. For instance in Figure 15-6, to map one pc0 onto another by following my rules, the first pc0 cannot stay; it must move via the shortest distance to get to the second pc0.

Figure 15-6, mapping a pc0 onto another pc0





The shortest distance is outlined by the dashed arrow, which directs this pc0 to circle around the entire clock to return to its original point. This full-circle progression projects the largest space of ic0 within this clock. Next, if we move one pc to another as shown previously in Figure 15-4, we derive the remaining six ics1–6 that have smaller spaces than ic0. Thus, with the aid of the clock schema, we can perceive ic1 and ic0 as the most and least dense ics identical to those defined in Example 15-1. Based on this spatial interpretation of the ics, I propose a harmonic measurement to test the density of a chord. Unlike Straus’s offset number, which relies on the cumbersome arithmetical process of fuzzy transformations to study the abstract spatial relationship among chords in terms of their sc representatives, my measurement is more contextsensitive. It contextualizes the ic-content of a chord, using a simple formula to study a more direct spatial relationship among chords with

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regard to their ordered pitch-class set (pcset).15 Figure 15-7 demonstrates my measurement.

Figure 15-7, contexualizing the ic-content of a harmony


The ics are formed by each pair of the adjacent voices. I call the VPicset— the voice-pair ic set. My premise is that if the VPicset predominantly comprises ic1, it results in a chromatic chord with a high harmonic density. Contrarily, if the VPicset predominantly comprises ic0, it makes a spacious chord with a low harmonic density. If the VPicset predominantly comprises ic6, it makes an even chord with a proportionally balanced density. To more accurately define the density of a chord, I propose a method that sums up all the members in the VPicset and averages the total, deriving a mean number called the average-VPicset.16 The smaller the average-VPicset, the higher density the chord. Also, if the average-VPicset is approximately close to 6, it represents an even chord. 17 There is, however, a crucial problem undermining this measurement. It works if and only if there is no ic0. I use Example 15-3 to demonstrate this problem, which is a brief passage from the piano accompaniment of the fourth song “Le Martin-Pêcheur” of Ravel’s Histoires Naturelles (mm. 18–19, see Example 15-3).

Example 15-3, Ravel, Le Martin-Pêcheur, piano part, mm. 18–19




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The right hand plays chord A in m. 18, which contains three pc4s separated by two octaves. This chord is repeated in m. 19 but in a lower register. Meanwhile, the left hand tenor part plays chord B, which unfolds a chromatic descending line from pc1 to pc0, pc11, and pc10. Example 154 uses ordered pcsets to represent chords A and B, in which the pcs are arranged from the lowest to the highest register.

4

1 ic0

4 ic0

ic1 0 ic1 11 ic1

ict nfl co

ict nfl co




4

Chromatic; Dense

chord B pcs

Spacious; Octaves

chord A pcs

10

Harmonic Density

High Density; Chromatic

Low Density; Spacious

average-VPicset:

(0+0)÷2= 0.00

(1+1+1)÷3= 1.00


Example 15-4, Ravel, Le Martin-Pêcheur, the average-VPticset

Theoretically, chord B composed of the chromatic pitches should be denser than chord A composed of octaves. However, their resultant average-VPicsets suggest an utterly conflicting and incorrect reading. The cause of this problem is the exact value of the number. Although the number 0 represents the most spacious ic, its exact value is smaller than the remaining ics1 to 6. During the process of averaging the sum of the ics, zero will significantly decrease the value of an average-VPicset. Thus, using zero to represent the most spacious ic turns out to be irrelevant for this harmonic measurement, and the final result cannot satisfy the notion of “the smaller the average-VPicset, the higher density the chord.” My solution to this problem is to revalue all the ics by subtracting one degree from each ic, mod 7 (see Example 15-5).


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even spacious

chromatic ics

1 -1

2 -1

3 -1

4 -1

5 -1

6 -1

0 -1

tics

0

1

2

3

4

5

6

density

high




low




Example 15-5, transformed ics (tics)


Since the seven sequential ics are repetitive and form a closed group, my revalued numbers must satisfy the same conditions. If subtracting one degree from the first ic1 derives the number of 0, the next six sequential numbers must range from 1 to 6. Hence, the last ic0 corresponds to the number 7 = 0 in mod 7, which, when 1 is subtracted, results in 6, representing the most spacious ic. Now the chromatic ic1 has the smallest value of 0, and the spacious ic0 has the largest value of 6. I call these revalued ics the “transformed ics” (tics). The greatest strength of these new revalued numbers is found in their ability to provide a more representative picture of “the smaller the tics, the more chromatic the ics.” Several ordered tics form a set called the voice-pair transformed ic set (VPticset). The average of a VPticset will be a number (average-VPticset) that defines the density of a chord, which ranges from 0.00 to 6.00.18 Within this range, the smaller the average-VPticset, the higher density the chord. Also, a chord can be regarded as even if its average-VPticset is approximately close to 5.00.19 Example 15-6 uses tics to replace ics in Example 15-4, and the average-VPticsets are 6.00 (chord A) and 0.00 (chord B).

Spaciousness or Evenness? A Theory of Harmonic Density chord A pcs

chord B pcs

4

1

ic0

tic6

301

ic1

tic0

ic1

tic0

ic1

tic0

0 4 11 ic0

tic6 10

4 average-VPticset:

(6+6)÷2= 6.00

(0+0+0)÷3= 0.00

Harmonic Density

Low Density; Spacious

High Density; Chromatic




Example 15-6, Ravel, Le Martin-Pêcheur, the average-VPticset


This result better interprets the density of chords A and B: four adjacent chromatic pcs are denser than those separated by two octaves. To accurately compare the densities among different chords, we must translate the ics into tics before any further computations. In the next section, I will show how this measurement works in music analyses, applying the average-VPticset to study pieces by Schoenberg, Berg, Crawford, and Kurtág.20 *** Examples 15-7 and 15-8 present chord progressions from the first piece of Arnold Schoenberg’s Drei Klavierstücke op. 11 (mm. 1–3) and the fourth song “Warm die Lüfte” from Alban Berg’s Vier Lieder op. 2 (mm. 20–21).




Example 15-7, Schoenberg, Drei Klavierstücke op. 11/1, mm. 1–3; the averageVPticsets

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Example 15-8, Berg, Vier Lieder op. 2/4, “Warm die Lüfte,” mm. 20–21; the average-VPticsets


They project the same type of texture, in which a principal melody is supported by a light chordal accompaniment. Despite this textural similarity, Schoenberg’s arrangement of his chromatic and spacious chords markedly differs from that of Berg’s. In Example 15-7, the left hand accompaniment contains two trichords, sc 3-5 and sc 3-3. Below each chord lies two items: the VPticset and the average-VPticset. The VPticset contains two successively vertical numbers of VPtics, and the bracketed number beneath the VPticset is the average-VPticset. According to the data, sc 3-5 is more spacious than sc 3-3. In other words, this harmonic progression moves from a lower-density to a higher-density chord. I define this type of progression as the process of “contracting the harmonic space.” Contrary to Example 15-7, the piano accompaniment in Example 15-8 projects a different type of harmonic progression. It contains two tetrachords sc 4-Z29 and sc 4-Z15, and this time sc 4-Z15 is more spacious than sc 4-Z29, resulting in a harmonic progression that moves from a higher-density to a lower-density chord. I define this progression as the process of “expanding the harmonic space.” Despite the similarity in texture, these two examples demonstrate two different underlying chord progressions.21

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The next analysis compares two passages from the third movement of Ruth Crawford’s String Quartet, mm. 20–29 and mm. 49–56 (see Examples 15-9 and 15-10).22


Example 15-9, Crawford, String Quartet, Mvt. III, mm. 20–29; the averageVPticsets







Example 15-10, Crawford, String Quartet, Mvt. III, mm. 49–56; the averageVPticsets


On the surface level, these two passages share several common contextual features, suggesting that they project a similar musical progression.23 They both begin with sc 4-1 followed by sc 4-13. These two scs are both articulated by the meter of 4/4
Although their final tetrachords are not the same— sc 4-1 in Example 15-9 and sc 4-Z15 in Example 15-10, they both last for a long duration of four measures, creating a static, frozen, and stable motion that suggests a sense of closure. In addition to these common




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contextual features, my application of the average-VPticset to analyze the chords in these two passages reveals a deeper relationship between Examples 15-9 and 15-10.
I assign each chord a number. Example 15-9 contains chords 1–6, and 7–11 are in Example 15-10. The average-VPticsets appear on the bottom of both examples. To more accurately describe the densities of these chords, I define a way to determine them to be either spacious or chromatic based on their corresponding average-VPticsets. Example 15-11 uses an enumerated line (on the top of this example) to represent all the average-VPticsets in Examples 15-9 and 15-10.
6 5

10 4

7

1

2

9

11

3

0.33

0.67

1.33

1.67

2.00

2.33

Chord #

8

Enumerated line Average-TVPicsets

0.61 0.62

0.33

0.90 0.91

1.19 1.20

1.48 1.49

1.77

1.78

2.06

2.07

2.33

Bi-directional line 0.28

0.28

0.28

Density: Extremely Chromatic Relatively Chromatic Slightly Chromatic

0.28

0.28

0.28

Neutral

Slightly Spacious

Relatively Spacious

0.26 Extremely Spacious

Example 15-11, defining the harmonic densities based on the average-VPticsets in Examples 15-9 and 15-10


The smallest average-VPticset 0.33 and the largest 2.33 frame it. On the bottom of Example 15-11 appears a bi-directional line spanning the same distance as the enumerated line above. This bi-directional line is equally divided into seven sections, and each one spans 0.28 degrees (with the exception of the one on the extreme right, which spans 0.26 degrees). Based on the notion of “the smaller the average-VPticset, the higher density the chord,” I define each section from left to right as describing a particular harmonic density— the extremely chromatic; the relatively chromatic; the slightly chromatic; the neutral; the slightly spacious; the relatively spacious; and the extremely spacious. Using these seven sections as the measuring references, I extend the divisions in the bi-directional line to meet the enumerated line by adding six vertical dotted lines. With these additional marks, the enumerated line is divided into seven sections identical to those seen in the bi-directional one. With the exception of the slightly chromatic section, every section contains one average-VPticset, which demonstrates a particular harmonic density— extremely chromatic: chord 7; relatively chromatic: chords 1, 5, 6; neutral: chords 2, 4, 10; slightly spacious: chord 9; relatively spacious: chord 11; and extremely spacious: chords 3, 8.




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To represent these average-VPticsets in a way that mimics the chronological sequence of chords, I use Graphs 15-1 and 15-2 to illustrate the relative degrees of the average-VPticsets in Examples 15-9 and 15-10, respectively.



Graph 15-1, densities of chords in Example 15-9 projected sequentially



Graph 15-2, densities of chords in Example 15-10 projected sequentially

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The vertical axis in the graphs represents the quantities of the averageVPticsets. The sc names and chord numbers appear on the horizontal axis. The dots, mid-graph, plot the average-VPticsets of the corresponding tetrachords along with their harmonic densities, defined in Example 15-11. Connecting the successive dots forms a graph that represents the progression of chord densities for its corresponding example. Graph 15-1 projects an arch representing the harmonic progression for Example 15-9. The first half of the arch (chords 1 to 3) forms a gradual ascent, which begins with the relatively chromatic chord and gradually rises to the extremely spacious chord by passing through a neutral chord. Reaching the highest value of the graph is only temporary, because just after chord 3 the arch immediately inverts the initial ascent and creates a symmetrical descent, which gradually reaches the relatively chromatic chord 5 by passing through a neutral chord 4. At the end, the arch maintains its motion by progressing to one more chord 6, whose averageVPticset is the same as that of chord 5. Based on the analysis of Graph 151, I define the arch to be the prolongation of a relatively chromatic chord density, as it begins and ends with relatively chromatic chords, reserving the neutral and the extremely spacious ones for the middle. Moving to Graph 15-2, the reader will notice two essential features that distinguish this graph from the previous one. First, each of the five dots in Graph 15-2 represents a chord with a different harmonic density— the extremely chromatic chord 7, the extremely spacious chord 8, the slightly spacious chord 9, the neutral chord 10, and the relatively spacious chord 11. Second, connecting all of these five successive dots forms a wave— a shape that differs markedly from the arch in Graph 15-1. It begins with a significantly direct ascent from a dot in the lowest range of the graph (chord 7) to that in the highest range (chord 8). The remainder of this figure becomes comparatively smoother, as it moderately descends to a neutral chord 10 by passing through a slightly spacious chord 9, and then rises again to a dot at the higher range of the graph— the relatively spacious chord 11. Through the above analysis, we understand that each graph contains a particular shape, which projects a markedly unique harmonic progression for its associated musical passage. Hence, I conclude that although Crawford uses several common contextual features to relate these two passages, she nonetheless carefully and deliberately distinguishes between them by superimposing them on two different harmonic progressions. My final analysis studies the relationship between text and harmony in György Kurtág’s chamber song “Intermezzo sul `An die aufgehende Sonne´” from his op. 37 Einige Sätze aus den Sudelbüchern Georg

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Christoph Lichtenbergs. It is written for soprano, trumpet, and horn (see Example 15-12).25 Part 1

Part 2

Soprano

, Trombe in C

Cor in Cor inFF (sounds as written)

Example 15-12, Kurtág, “Intermezzo sul `An die aufgehende Sonne´”


Among these three voices, the soprano is the principal melody and is imitated by the other two instruments with some slight variations. All the pitches in the horn have been transposed down a perfect fifth, representing their corresponding concert pitches. The text, Georg Lichtenberg’s aphorism, is a sentence comprised of two clauses separated by a comma— “Was hilft aller Sonnenaufgang, wenn wir nicht aufstehn.” I use my own translation for the following discussion— “What is the point of the sunrise, if no one gets up.” Although this aphorism contains only a few words, it is noteworthy in that it possesses two contrasting intonations. Phonologically, while the first clause projects a rising intonation that necessarily conveys a sense of instability, the second projects a falling intonation, which necessarily conveys a sense of stability that closes the whole aphorism. Musically, to accommodate such a succinct aphorism, Kurtág sets it to a brief, simple melody sung by a soprano. Observe the melodic contour of the soprano voice, it generally progresses in an upward direction before the rest mark “ ” in m. 3, and then in a downward direction.26 Comparing the contour of the soprano voice and the text’s intonation, I find a significant parallel between these two elements— rising intonation versus upward contour, and falling intonation versus downward contour. This comparison reveals how Kurtág composes a melody by contextually mimicking and integrating the changing intonation of the text. Besides the melodic contour, my project is to study whether Kurtág also uses chromatic and spacious chords to compose a harmonic progression complementing the text’s two different intonations. According to the comma in the text and the breath mark in measure 3, I divide the song into two parts.27 Here I need to explain my method of




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chord segmentation. Example 15-13 begins by marking either similar or identical musical figures that exist among the three imitative voices.








Example 15-13, Kurtág, “Intermezzo,” Units An, Bn, Cn, and Dn


I call these marked musical figures “units.” To derive a harmony, I combine the simultaneous units across all three voices, creating a chord containing a particular combination of different units. In Example 15-13, each unit is framed by a different line-type (solid/dashed) box or oval, which is further marked with one of the capital letters A (solid box), B (dashed box), C (solid oval), and D (dashed oval). To identify the units with the same letter names among these three voices, I add subscripts i for the soprano, ii for the trumpet, and iii for the horn. Example 15-14 uses a larger box to combine the simultaneous units across the three voices to create a chord.

Example 15-14, Kurtág, “Intermezzo,” harmonic segmentations




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Part 1 contains chords 1–4, and chords 5–7 appear in Part 2. To add clarity to my chord segmentations, Example 15-15 removes all the smaller boxes and ovals.




Example 15-15, Kurtág, “Intermezzo,” the average-VPticsets


The average-VPticsets appear on the bottom of this example. Similar to my previous analysis of Crawford’s work, Example 15-16 uses an enumerated and bi-directional line to define the harmonic densities.
7

4

1.00

1.50

Chord #

6

2

5

3

1

3.50

4.00

Enumerated line average-VPticsets

1.80

1.59 1.60

1.00

2.67

2.19 2.20

2.80

2.79 2.80

3.39 3.40

4.00

Bi-directional line 0.59

0.59

Extremely Chromatic Relatively Chromatic

0.59

0.59

0.60

Neutral

Relatively Spacious

Extremely Spacious

Example 15-16, the harmonic densities based on the average-VPticsets in Example 15-15


Chords 4 and 7 are extremely chromatic; chord 6 is relatively chromatic; chord 2 is neutral; chord 5 is relatively spacious; and chords 1 and 3 are extremely spacious. To represent the seven average-VPticsets of Example 15-16 in a way that mimics how the harmonies flow sequentially, I use a graph format (Graph 15-3) to project the chord progression in Kurtág’s “Intermezzo.”





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Figure A 4.00 extremely spacious

4.00

3.50 extremely spacious relatively spacious

3.00

2.80 relatively chromatic

2.67

1.80

neutral

2.00

1.50

1.00

extremely chromatic

1.00 extremely chromatic

0.00 1

2

3

4

5

6

7

Graph 15-3,
graphical representation of chord progression in Example 15-15: Figure A


I call the image in this graph “Figure A.” It starts with the highest dot, passes through another five dots and arrives at the lowest one. Considering the overall contour of this figure, it resembles a large-scale descent from the highest to the lowest dot in the graph, projecting a harmonic progression that moves from an extremely spacious to an extremely chromatic chord. If we perceive this progression as the process of contracting or expanding a chord space, it basically compresses the space until the last chord— the least spacious one that completes the process of contraction.28 After discussing the large-scale contour of Figure A in Graph 15-3, I turn my focus to the smaller details of this figure. Graph 15-4 separates Figure A into two smaller figures representing Parts 1 and 2— Figures B and C.





Spaciousness or Evenness? A Theory of Harmonic Density

Part 2/ Figure C

Part 1/ Figure B 4.00

311

4.00

extremely spacious

3.50 extremely spacious 3.00

2.80 relatively spacious 2.67

1.80

neutral

2.00

relatively chromatic

1.50

extremely chromatic

1.00

1.00 extremely chromatic

0.00 1

2

3

4

5

6

7

Graph 15-4, separating Graph 15-3 into Figures B and C based on Parts 1 and 2


At first glance, both of these two new figures project a similar harmonic progression; they both begin and end with the largest and the smallest average-VPticsets within their corresponding parts. While Figure B moves from the extremely spacious chord 1 to the extremely chromatic chord 4, Figure C moves from a relatively spacious chord 5 to the extremely chromatic chord 7. Thus, both Figures B and C form two generallydescending contours, representing a similar type of large-scale harmonic progression from a spacious to a chromatic chord. But when I look more closely at the details within Graph 15-4, I notice several crucial differences distinguishing Figures B and C from one another. Strikingly, just after passing a neutral chord 2, Figure B starts to wander and even moves up to chord 3 in the higher range of the graph. This salient attempt to return to the higher range of the graph significantly deviates from the overall descent, resulting in an indecisive, hesitant, and circuitous figure representing the chord progression for the first part of Kurtág’s song. Contrarily, Figure C is far more direct. It starts with a relatively spacious chord 5, which descends 1.00 degrees to a relatively chromatic chord 6. Then the progression continues by descending another 0.80 degrees, arriving at the last chord of this song. The resultant figure is a nearly straight and undeviating slope. To summarize my analysis of Graph 15-4, I find the processes of completing the descents in Figures B and C to be essentially different from one another. These differences




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would be more analytically significant if we consider them in connection with the text’s intonation. As previously discussed, each of the two clauses in the text, “What is the point of the sunrise, if no one gets up,” carries a distinctive and contrasting intonation. Based on my analysis of Graph 15-4, I find that besides the melodic contour of the soprano voice, Kurtág also uses chords to reflect and represent the text’s changing intonation. For instance, while the first clause contains a rising intonation that expresses a sense of instability, its associated Figure B is a circuitous descending line, demonstrating an indirect progression from the extremely spacious to the extremely chromatic chord. Contrarily, while the second clause changes to a falling intonation that expresses a sense of stability, its associated Figure C imitates this change of intonation and turns into an almost straight slope, demonstrating a direct progression from a relatively spacious to an extremely chromatic chord. Thus, compared with Graph 15-3, my analysis of Graph 15-4 shows a deeper and closer relationship between the text’s intonation and its accompanying chord progression.29 *** The above analyses illustrate the practical advantage of the averageVPticset as realized in the compositions by Schoenberg, Berg, Crawford, and Kurtág. Finally, I would like to conclude my chapter with several suggestions to theorists who are interested in the further development of my chord measurement. Initially, my topic was concerned with the necessity of separating the sense of harmonic evenness from that of spaciousness. Despite the fact that the separation successfully assists us in more accurately perceiving the sense of evenness, its probability of occurrence in actual pieces of music is quite low. For instance, none of the above four analyses include any even chords, or any nearly-even chords with an average-VPticset 5.00. Though they do appear in music, we still need the existence of another more spacious chord with the averageVPticset 6.00 as a reference to reinforce the sense of evenness. Otherwise, the existence of an even chord by itself can be merely interpreted as a spacious chord. For example, if we have five average-VPticsets 1.50, 2.75, 3.00, 4.67, and 5.00, based on the notion of “the smaller the averageVPticset, the higher density the chord,” the 5.00 here would more likely be understood as the most spacious chord instead of an even one, for its value is larger than the other four. Only when we add another average-VPticset 6.00 can we interpret the 5.00 as an even chord. Considering this issue, my next project will further refine my current measurement, seeking a more

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accommodating approach that takes greater account of the information given by the musical context, observing the contextually-even chord appearing between the extremes of the chromatic and spacious chords within any given musical passage.

Notes 1

The author wishes to thank Martha Hyde, Philip Stoecker, Yu-Lin Hsiao, and Scott Murphy, who provided valuable advice during the various stages of developing and shaping his methodology. He would also like to thank Jack Boss, Russell Knight, Kathleen Marien, and Patrick Sheley, who offered invaluable assistance and knowledge throughout the writing and editing process. 2 For a detailed summary and comparison of different sc similarity measurements, see Eric Isaacson, “Similarity of Interval-Class Content between Pitch-Class Sets: The IcVSIM Relation,” Journal of Music Theory 34/1 (1990): 1–29, Michael Buchler, “Relative Saturation of Interval and Set Classes: A New Model for Understanding Pcset Complementation and Resemblance,” Journal of Music Theory 45/2 (2001): 263–343, and Ian Quinn, “General Equal-Tempered Harmony (Introduction and Part I),” Perspectives of New Music 44/2 (2006): 114–58 and “General Equal-Tempered Harmony: Parts 2 and 3,” Perspectives of New Music 45/1 (2007): 4–63, among others. Additionally, for critique and discussion about the overall concept of similarity in music set theory (such as sc similarity and contour similarity), see Isaacson, “Issues in the Study of Similarity in Atonal Music,” Music Theory Online 2/7 (November 1996) and Quinn, “Listening to Similarity Relations,” Perspectives of New Music 39/2 (2001): 108–58. 3 In his treatise, The Craft of Musical Composition, Book I: Theory, trans. Arthur Mendel (New York: Schott Music, 1984 [1937]), Paul Hindemith categorizes chords into six types. Despite the fact that Hindemith uses the terms “stability and instability” instead of “consonance and dissonance” to discuss his chord types, he does mention consonance and dissonance during his discussion of intervals prior to his chord categorization. Although Hindemith proposes different terms to define the quality of interval and chord, there is, nevertheless, a close correlation between these two elements. Each chord type reflects a particular level of stability based on its root and the number of dissonant intervals. For instance, if a chord contains a determinate root and does not include any dissonant intervals, it represents the most stable harmony (his chord type 1). Contrarily, if a chord lacks a determinate root and includes the dissonant interval of the tritone, it represents the most unstable harmony (his chord type 6). Perle’s concept of dissonance and consonance is slightly different from our traditional sense. He regards consonance to be a dyad chord featuring a symmetrically balanced structure, and the dissonances are the passing tones that fill in the space between any two consecutive consonant chords. For detailed discussion, see his Twelve-Tone Tonality, 2nd ed. (Berkeley and Los Angeles: University of California Press,

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1996), 229–34. Moreover, beyond the theoretical speculation about the quality of a chord, David Huron, in his “Interval-Class Content in Equally Tempered PitchClass Sets,” Music Perception 11 (1994): 289–305, uses cognitive science to study our human perception of the tonal consonance of a sc (tonal here does not mean tonality; it simply refers to pitch). 4 Joseph Straus, “Voice Leading in Set-Class Space,” Journal of Music Theory 49/1 (2005): 45–108. In addition, Straus also discusses his fuzzy transformational voice leading in other three publications: “Voice Leading in Atonal Music,” in Music Theory in Concept and Practice, eds. James Baker, David Beach, and Jonathan Bernard, 237–74 (Rochester: University of Rochester Press, 1997); “Uniformity, Balance, and Smoothness in Atonal Voice Leading,” Music Theory Spectrum 25/2 (2003): 305–52; and “Some Additional Relationships: Voice Leading,” in Introduction to Post-Tonal Theory, 3rd ed., 107–10 (Upper Saddle River, NJ: Pearson Education, 2005). Hereafter, all of my citations are from his “Voice Leading in Set-Class Space.” 5 In the literature of the related topic, we can also find other theorists who study the density of a chord, such as Michael Klein, “A Theoretical Study of the Late Music of Witold Lutosławski: New Interactions of Pitch, Rhythm, and Form” (Ph.D. diss., the University at Buffalo: 1995), and “Texture, Register, and Their Formal Roles in the Music of Witold Lutosławski,” Indiana Theory Review 20/1 (1999): 37–70, and Quinn, “General Equal-Tempered Harmony: Parts 2 and 3.” 6 Straus spends a small portion of his article, “Voice Leading in Set Class Space” (p. 73–83), arguing that his definition of harmonic quality shares a close correspondence with the traditional notions of consonance and dissonance. But beyond this portion of his paper, the majority of the discussion is still concerned with the density of a chord, for 1) Straus consistently uses the term “degree of chromaticness” to define a chord in his analyses, and 2) his proposal of the Law of Atonal Harmony stresses the overall shift in the different levels of density outlined by a succession of chords, which asserts that “harmonies [in atonal music] tend to move away from a state of chromaticness and toward a state of relative dispersion” (p. 83). In this regard, my research takes on his quality of harmonic density. 7 {0,1,2} in sc 3-1 are generated by ic1: 0 1 2, and {0,4,8} in sc 4-12 are generated by ic4: 0 4 8. 8 My article, “An Issue between Contemporary Theory and Modern Compositional Practice: A Study of Joseph Straus’s Laws of Atonal Voice Leading and Harmony Using Webern’s Opus 12/2 and Crawford’s String Quartet Mvt. 3,” Revista Vórtex 1/2 (2013): 1–29, thoroughly studies Straus’s fuzzy transformational voice leading and uses Webern’s Op. 12/2 and Crawford’s String Quartet as test cases to evaluate the practical application of his theory. 9 Straus also measures these six scs against sc 2-6 (see Figure 15-N1), and the resultant offset numbers reverse those shown in Figure 15-2. Thus, as dyads move away from sc 2-1, their corresponding offset numbers become smaller. Although here each sc always has two complementary offset numbers creating a sum of 5,

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this consistency does not apply to scs that have a cardinality larger than three. Straus refers to these inconsistencies as “rogues” (p. 68). To avoid this problem, Straus only measures scs against the most chromatic sc after his Figure 15.

Figure 15-N1, from Straus, “Voice leading in Set Class Space,” p. 68 10 Straus, p. 67. 11 As pointed out in note no. 6, Straus also uses consonance and dissonance to describe the six ics. He first observes that “the traditionally dissonant harmonies tend to be the most chromatic ones…[and] the traditionally consonant or stable harmonies are always among the least chromatic” (p. 73). He relies on these traditional notions of consonant and dissonant harmonies to define the qualities of the six ics1–6 in Figure 15-3. Comparing the opposing extremes of the ics, since ic1 and ic6 are respectively the most and least chromatic ics, they accommodate the most dissonant and the most consonant ics. Thus, the range of offset numbers 0–5 in Figure 15-3 outlines the two extremes of the most consonant and the most dissonant ics. Within this range, the smaller the offset number, the more dissonant the ic; the larger the offset number, the more consonant the ic. Two questions beg further study in relation to the quality of an ic. The first is the ambiguous quality of ic6. In his article, Straus acknowledges that this ic— the tritone— is “obviously not a traditional consonance,” but “the traditionally most consonant dyad, 2-5 [i.e., ic5], is only one degree of offset away” (p. 73). Here, ic6 bears a conflicting definition. Theoretically, it has the largest offset number, which seems to imply the notion that the tritone is not only a consonant interval but also the most consonant one. Practically, however, it is not a consonance. In this sense, the premise of “the larger the offset number, the more consonant the ic” becomes less convincing and legitimate. (The quality of the tritone in post-tonal music has been challenged by many scholars, who have debated this issue and attempted to seek a theoretically convincing way to define the tritone as either a consonance or

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dissonance. For instance, Charles Seeger, “Tradition and Experiment in (the new) Music,” in Studies in Musicology II: 1929–1979, ed. Ann M. Pescatello (Berkeley and Los Angeles: University of California Press, 1994), hears it sound “more consonant chordally than melodically” (p. 40); Ernst Krenek, Studies in Counterpoint: Based on the Twelve-Tone Technique (New York: G. Schirmer Inc, 1940), assumes that its “character of a consonance or of a dissonance depends on the third tone added” (p. 20); and Hindemith, The Craft of Musical Composition, even more puzzlingly, argues the tritone to be a “dissonance” but not “cacophony” (p. 85). The above three examples juxtapose contrasting interpretations concerning the quality of the tritone and reveal representative kinds of disagreements about this interval among post-tonal composers.) The second question concerns the absence of ic0 in Figure 15-3, which represents both the octave and unison. If we were to consider this ic as well, what would the result be? Would it still place the ic6 as the most consonant interval? If so, wouldn’t this essentially conflict with the fact that the octave has always been considered to be the most perfect and consonant interval? If not, would ic6 still represent a consonance? 12 The detailed and classic study of evenness appears in John Clough and Jack Douthett’s “Maximally Even Sets,” Journal of Music Theory 35/1–2 (1991): 93– 173. Like my Figure 15-4, Clough and Douthett use the clock image as a schema to represent their 12-pc universe. Within this clock, if a set’s pcs are distributed as evenly as possible around the circle, it conforms to what they describe as a maximally even set. Among all the scs from cardinality two to ten, we can always find “one” maximally even set in each sc group— such as the tritone (sc 2-6), augmented triad (sc 3-12), and diminished seventh chord (sc 4-28), among others. Note that Clough/Douthett and Straus’s theories disagree with regard to the presence of exactly one maximally even set in each sc group. Both theories claim that sc 3-12 is the only maximally even sc among all the trichordal scs, but once we get to chords with larger cardinalities, they have significantly different views on the number of even scs. For instance, among all the tetrachordal scs, Clough and Douthett believe sc 4-28 is the only and maximally even sc, while we find two more even scs (4-9 and 4-25) alongside sc 4-28 in Straus. For Straus, these three sets belong to the most even tetrachords, because they receive the same largest offset number 8 measured against sc 4-1. Regarding this disagreement, Straus admits that if we measure sc 4-9 and sc 4-25 against Clough/Douthett’s maximally even sc 4-28, we will find both sc 4-9 and sc 4-25 are not nearly as even as sc 4-28. This is a situation that Straus refers to as “rogue” pointed out previously in note no. 9. Confronted with this anomaly, Straus suggests the reader to focus on the quality of chromaticness between sc 4-9/sc 4-25 and sc 4-1, while that of evenness is “less-defined,” which is “casually” understood among sc 4-9, sc 4-25, and sc 4-28 (Straus, p. 71). 13 In Straus’s article, he also measures offset numbers among scs with different cardinalities (see his Figure 17, p. 72).

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14 In the history of Western music theory, there are other theorists who do not hear the unison as forming an interval. For instance, Gioseffo Zarlino, in his book The Art of Counterpoint, translated by Guy A. Marco and Claude V. Palisca (New Haven: Yale University Press, 1968), regards the unison to be the beginning of an interval, but not an interval in itself. In his discussion, he uses the geometric elements of “point” and “line” as the analogy to compare the relationship between a unison and other intervals. He says:

A point is the beginning of a line, although it is not itself a line. But a line is not composed of points, since a point has no length, width, or depth that can be extended, or joined to another point. So a unison is only the beginning of consonance or interval; it is neither consonance nor interval, for like the point it is incapable of extension (p. 24; emphasis mine). Besides Zarlino, Jean-Philippe Rameau in his Traité de l’Harmonie, translated by Phillip Gossett (New York: Dover Publications, 1984) also rejects the unison as an interval, for it is “only a single sound which may be produced by several voices or by several instruments” (p. 8; emphasis mine). This way of hearing the unison produced by different voices/instruments is similar to my theory, which defines it as a tone repetition. 15 In an ordered pc set, I arrange the pcs according to musical context. They may be arranged from low to high registers, from bass to treble instruments, from weak to strong dynamics, or other contextual means. In this regard, my measurement shows a great flexibility that can study the density of a chord from multi-faceted presentations of contextualized ic-contents, providing the reader with a more insightful analytical interpretation that meets the needs of various musical contexts. 16 In Figure 15-7, the average-VPicset is (x+y+z)÷3. 17 That means the chord is primarily composed of ic6, which is the only even ic in Example 15-1. 18 If a VPticset is only composed of tic0, summing up all the tics and averaging the total derives the average-VPticset of 0.00, which corresponds to the smallest number and represents the highest harmonic density. Contrarily, if a VPticset is only composed of tic6, summing up all the tics and averaging the total derives the average-VPticset of 6.00, which corresponds to the largest number and represents the lowest harmonic density. Since the smallest and the largest average-VPticsets are 0.00 and 6.00, connecting these two numbers with each other then forms a limited range that covers all the possible average-VPticsets derived from the application of my harmonic measurement. 19 That means a chord is primarily composed of tic5, which is the only even tic in Example 15-5. 20 This note provides the reader with other alternative theories that also use the concept of space to study chords in 20th- and 21st-century music. In Klein’s 1995 doctoral dissertation “A Theoretical Study of the Late Music of Witold

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Lutosławski” and 1999 article “Texture, Register,” he proposes a theory called the compression of a harmonic aggregate. Unlike my average-VPticset, Klein uses pitch-space instead of pitch-class space. In pitch-space, there is no octave equivalence. The different registral pitches appearing on the same beat form a harmonic aggregate. In this aggregate, the space— the total number of pitches— is called a field (f). Density (d) refers to the cardinality (the number of different pitches) of the harmonic aggregate. Dividing density by field derives compression (c), which is a number ranging from 1.00 to 0.01 (the formula is c = d/f). For Klein, this resulting number expresses “how tightly packed the voices are within a textural field” (“Texture, Register,” p. 44). Within this arrangement, the larger the number, the more compressed the aggregate. Another more recent approach appears in Quinn’s “General Equal-Tempered Harmony: Parts 2 and 3”— his Lw and F (12,1). The “F” is the abbreviation for Fourier space. The two numbers, 12 and 1, respectively represent the twelve chromatic pcs and the arrangement of these twelve pcs (in this case, 1 means that the twelve pcs are generated by ic1, resulting in an ordering from pc0 to pc11). The Lw, which is short for lewin, is a measurement that tests the degree of density of a chord in relation to this chromatic scale F (12, 1). The larger the Lw, the stronger association between a chord and the presentation of F (12, 1), and the more chromatic that chord. In addition to Klein’s compression and Quinn’s F (12,1), it is worth mentioning two other harmonic theories that are also concerned with the space of a chord. However, instead of studying how chords contract and expand their densities, these two theories focus on different issues that are associated with the space of a chord. The first appears in Alan Chapman’s 1981 article (“Some Intervallic Aspects of Pitch-Class Set Relations,” Journal of Music Theory 25/2: 275–90), in which his voice pairs interval set (abbr. VP) gauges the ordered pc interval between each pair of adjacent voices in one chord. If chords whose VPs share the same combination but with different orderings of the ordered pitch-class intervals, they create VP-related sets. Importantly, throughout his discussion, Chapman is primarily concerned with the space produced by VPs or VP-related sets, for different orderings of a VP can result in different scs. The second appears in Jonathan Bernard’s 1981 article (“Pitch/Register in the Music of Edgard Varèse,” Music Theory Spectrum 3: 1–25). In parallel to Klein’s theory, Bernard defines the space of an interval similar to Klein’s field. However, different from Klein’s f, Bernard gauges the total number of semitones— not the pitches— from the lowest to the highest pitches (i.e., unordered pitch intervals). Based on his perception of an interval, Bernard studies the structural similarity between different chords in terms of the arrangements of their unordered pitch intervals. The analytical techniques he proposes include symmetry, mirror symmetry, parallel symmetry, projection, partial projection, rotation, expansion, and contraction. Also, grounded in the same definition of the space of an interval, Bernard in his 1994 article (“Voice Leading as a Spatial Function in the Music of

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Ligeti,” Music Analysis 13: 227–53) extends his discussion to include the subject of voice leading, examining how the formation of the voice leading is reinforced not by “literal, registral proximity” but by the spatial context of the “absolute sizes of intervals” (p. 230) in Ligeti’s Lux aeterna and his Second String Quartet, Mvt. III. 21 The definitions of expanding and contracting chord space will be crucial for my later Kurtág analysis. 22 Note that voice crossings constantly appear throughout Examples 15-9 and 1510. (Voice crossing here refers to a pair of adjacent string instruments whose registral orderings do not follow the traditional, standard string quartet setting, in which Vln I, Vln II, Vla, and Vc, respectively, project the soprano, alto, tenor, and bass voices.) In her article “The Question of Climax in Ruth Crawford’s String Quartet, Mvt. 3,” in Concert Music, Rock, and Jazz since 1945, eds. Elizabeth West Marvin and Richard Hermann (Rochester: University of Rochester Press, 1995), Ellie Hisama uses the term “degree of twist” to define the number of voice crossings within a chord in this particular movement (p. 298). For instance, chord 1 in Example 15-9 has a degree of twist 2— Vc over Vla and Vln II over Vln I. Based on her analysis of the degree of twist, Hisama discovers multiple climaxes achieved by the gradual progressions moving toward and then away from an exceedingly twisted texture, which do not coincide with the registral and dynamic climax at m. 75. For Hisama, the highly twisted texture represents what she believes to be a “feminist” climax (p. 305), which is opposed to the register and dynamics representing a more contextual, traditional, and “masculine” form of climax in a composition. (Additionally, see Edward Gollin, “Form, Transformation and Climax in Ruth Crawford Seeger’s String Quartet, Mvmt. 3,” Mathematics and Computation in Music 37 [2009]: 340–46, where he extends Hisama’s study by applying a Cayley graph to analyze how the registral permutations of the four string instruments create a transformational network, and further uses his results to support Hisama’s view of feminist climax in Crawford’s String Quartet, Mvt. III.) 23 Straus also provides the harmonic analysis of this entire movement (see his The Music of Ruth Crawford Seeger, Cambridge: Cambridge University Press, 1995). For most of this particular movement, the adjacent tetrachords differ from one to another by the movement of a single pitch from the first to the next chord, with the rest of the pitches remaining fixed. Connecting these moving pitches forms the main melody (p. 170). My chordal segmentations of Crawford’s examples are all based on this main melody: a new pitch creates a new chord. For instance the first two tetrachords sc 4-1 and sc 4-13 in Example 15-9 have three invariant pcs, {0,1,3}, while the variant ones appear in the first violin (pc2 in sc 4-1 and pc6 in sc 4-13). Tracing the different pcs in each adjacent pair of chords, the main melody is composed of pc2 (Vln I, sc 4-1), pc6 (Vln I, sc 4-13), pc3 (Vc, sc 4-12), pc11 (Vc, sc 4-18), pc2 (Vln I, sc 4-3), and pc1 (Vc, sc 4-1). The main melody of this movement has been thoroughly analyzed by Straus in terms of its structure, as well as its pitch and duration contours. Next, he begins to

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examine the vertical dimension— the harmony. But rather than emphasizing the chord-to-chord succession, Straus only discusses the higher level of the harmonic progression. For instance, according to Straus, the initial and final tetrachords in my Example 15-9 (chord numbers 1 and 6) are T4-related to the first tetrachord in my Example 15-10 (chord number 7). These three tetrachords form a large-scale prolongation of sc 4-1. Although his examination shows a deeper structural level of harmonies, a detailed discussion of the local chordal progression is lost. Complementary to Straus’s analysis, I examine each passage in terms of its harmonic progression on a foreground level, and further compare the differences and similarities between Examples 15-9 and 15-10. 24 Crawford repeatedly alternates the meters of 3/4, 4/4, and 5/4 in this movement. 25 It is worth mentioning that a few years ago the Italian music theory journal Gruppo di Analisi e Teoria Musicale published a special issue concentrating on Kurtág’s music. This issue contains articles by Friedemann Sallis, Antonio Rostagno, Egidio Pozzi, and Mario Baroni. For more information, see Gruppo di Analisi e Teoria Musicale XVI/1 (2010): 7–122. 26 Since the soprano is the leading, principal voice among the three imitative voices, I regard it as the primary melody accompanied by two other brass instruments and focus on its melodic contour. 27 Regarding the issue of formal division, Patricia Howland, in her recent article “Formal Structures in Post-Tonal Music,” Music Theory Spectrum 37/1 (2015): 71–97, integrates various parametric elements such as rhythm, temporal density, texture, and contour to study formal structure in twentieth and twenty-first century music. She systematically categorizes five types of scenarios that describe boundaries among different formal segments: tension/release, departure/return, symmetric, directional, and steady-state. Since my paper briefly touches on the issue of form here in Kurtág’s song, for a deeper and more detailed discussion please refer to Howland’s article. 28 As Rachel Beckles Willson would likely observe (Ligeti, Kurtág, and Hungarian Music During the Cold War [Cambridge: Cambridge University Press, 2007]), Kurtág’s use of perfect fifths in the soprano’s first phrase invokes the “innovation of God,” which can be regarded as the depiction of the word “sunrise” in the text, while his use of a pitch C#4 that is chromatic and dissonant to the whole tone collections {0, 2, 8, T} in the soprano’s second phrase (I define these pitches as “all-but-one whole-tone collections”) suggests “sin” and “death,” which might be seen as passing judgment on those who do not get up early (p. 107–11). Combining her observation with my analysis, I find that not only is there a phonological change between the two phrases (i.e., pitch contour), there is also a semiotic one (i.e., the intervals). But how does my analysis of Graph 15-3 support this semiotic articulation? Beckles Willson also mentions Kurtág’s use of chords to project what she refers to as “dark, dirty” and “light, clean” sonorities (p. 106). She uses “space” to describe each of these two types of sonorities. While the first sounds “closed,” the second sounds “open.” If we replace Beckles Willson’s terms with

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my words, the open and closed sonorities would correspond to spacious and chromatic chords, respectively. Then the overall chord progression from the spacious to the chromatic in Graph 15-3 beautifully supports Kurtág’s semiotic design, in that the spacious chord (which signifies the topic “light, clean”) along with the perfect fifths delineate the picture of the “sunrise,” and the chromatic chord (which signifies the topic “dark, dirty”) along with the “all-but-one wholetone collections” delineate the picture of “no one gets up.” 29 Additionally, in my forthcoming article “A New Similarity Measurement of Pitch Contour for Analyzing 20th- and 21st-Century Music: The Minimally Divergent Contour Network,” Indiana Theory Review 31, I study Kurtág’s “Intermezzo,” but from a melodic contour perspective, and find another intimate relationship between contour similarities among the three imitative voices and text.

CONTRIBUTORS

Sara Bakker holds the Ph.D. in music theory from Indiana University, where she taught music majors and minors. She currently teaches at Utah State University in Logan, UT. Her dissertation on rhythmic phase shifting in the music of György Ligeti shows how subtle differences in the phase relationship result in pieces with drastically different formal and aesthetic outcomes. These analyses address ways to achieve closure in repetition-intensive repertoire, formal compression and expansion through varied repetition, and recursive structures at multiple organizational levels. This chapter and her forthcoming chapter in György Ligeti Symposium (forthcoming from University of Rochester Press) expand on this research. Dr. Bakker has also presented research on rhythm and meter, including rhythmic aspects of Hungarian text-setting (Music Theory Midwest 2008), formal features of Baroque unmeasured preludes (Dutch-Flemish Society for Music Theory 2009), and teaching rhythm and form (Society for Music Theory 2009). She has served on the editorial board for Indiana Theory Review (2006–11), is a reviewer for Oxford University Press, and currently serves on the Society for Music Theory’s Committee on the Status of Women (2014–16). Jack Boss holds the Ph.D. in music from Yale University, where he studied with Allen Forte, Claude Palisca and David Lewin. He is presently Professor of Music Theory and Composition at the University of Oregon. He has also taught at Brigham Young University, Ball State University and Yale. His monograph, Schoenberg’s Twelve-Tone Music: Symmetry and the Musical Idea, was published by Cambridge University Press in October 2014, and was the winner of the Wallace Berry Award from the Society for Music Theory. He has also co-edited two essay collections for Cambridge Scholars Publishing: Musical Currents from the Left Coast (2008) and Analyzing the Music of Living Composers (and Others) (2013). He has published a number of influential articles on Schoenberg’s, Beethoven’s and Bernard Rands’s music in Music Theory Spectrum, Journal of Music Theory, Perspectives of New Music, Intégral, Music

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Theory Online and Gamut. His book reviews can be found in the Journal of Music Theory, Music Theory Online and Notes. Dr. Boss has given numerous conference presentations on Schoenberg’s music and related topics in the United States (AMS/SMT and CMS, as well as many of the regional societies), Canada, England (SMA), Ireland (Dublin International Conference on Music Analysis) and Belgium (Dutch-Flemish Society for Music Theory). He served from 1989-91 as Reviews Editor, Associate Editor and Editor in Chief of the Journal of Music Theory, from 2000-05 as Reviews Editor for Music Theory Online, and from 2005-10 on the Editorial Review Board of the Journal of Music Theory Pedagogy. He was elected President of the West Coast Conference of Music Theory and Analysis in 2003. Timothy Chenette has a Ph.D. in music theory from Indiana University, where his dissertation explored transformational approaches to voice leading and harmony in late sixteenth-century vocal music. He currently serves as Assistant Professor and head of the music theory area at Utah State University; he taught previously at the University of MassachusettsAmherst and Indiana University. In addition to his dissertation topic, Chenette has also done extensive research on syncopation and meter in late fourteenth-century music and music theory pedagogy. In 2014, he founded an interest group within the Society for Music Theory dedicated to furthering the analytical study of early music, a group he now chairs. He has published in Music Theory Online and presented at Music Theory Midwest, the New England Conference of Music Theorists, the West Coast Conference of Music Theory and Analysis, the Gesualdo 400th Anniversary Conference in York, England, and several times at the Society for Music Theory annual conference. He also co-authored an article on writing in music theory curricula for Engaging Students: Essays in Music Pedagogy, vol. 2. Laura Emmery holds a Ph.D. in Music Theory from University of California, Santa Barbara and a Master of Music Degree in Theoretical Studies from the New England Conservatory. Under the guidance of Patricia Hall and Pieter van den Toorn, her dissertation on Elliott Carter’s string quartets incorporates sketch study in tracking Carter’s historical evolution and compositional process. Receiving the Paul Sacher Stiftung Scholarship, Emmery spent eight months in Basel working with the original sources. She is also the recipient of numerous other fellowships and grants, including the Arizona State University School Research Grant, Emory University Research Grant, UCSB Affiliates Graduate Dissertation

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Fellowship, Humanities and Social Sciences Research Grant, and Albert and Elaine Borchard European Studies Fellowship. Emmery is currently Assistant Professor of Music Theory at Arizona State University, Herberger Institute for Design and the Arts, School of Music. Her previous teaching posts include Emory University and University of California, Santa Barbara. Laura Emmery has presented her work in national and international conferences, including Tracking the Creative Process in Music, Organized Time—Fifteenth Annual Congress of the Gesellschaft für Musiktheorie, Society for Music Theory, Music Theory Midwest, West Coast Conference of Music Theory and Analysis, Music Theory Society of New York State, European Music Analysis Conference, Sacher Perspectives: Musicology at the Paul Sacher Foundation—New Directions in Source Study, Paul Sacher Stiftung Colloquium, and Temporality: Issues of Change and Stasis in Music. She has published articles on Elliott Carter's music in Twentieth-Century Music, Tempo, Sonus, and Mitteilungen der Paul Sacher. Matthew Ferrandino earned his MA degree in Music Theory from the University of Oregon in 2015, with a thesis on “What to Listen For In Zappa: Philosophy, Structure, and Allusion in Frank Zappa's Works.” Matthew has presented at the West Coast Conference for Music Theory and Analysis and the Western University Graduate Symposium on Music. He holds a BM in Composition from the Hartt School of Music and a MM in Composition from SUNY Fredonia. Susan de Ghizé is an Associate Professor of Music Theory at the University of Texas at Rio Grande Valley. She received her Ph.D. from the University of California at Santa Barbara and her B.A. from the University of California at Berkeley. Previously, de Ghizé taught at the University of Denver, Northeastern University, and the National University of Singapore. She has presented at national and international conferences on topics varying from traditional forms in Leos Janačék’s Second String Quartet to utilizing technology in the music theory classroom. She has also published articles about rhythmically developing variations in Brahms’s chamber works and applying Hauptmann’s theory of meter to Mozart's Piano Sonata K. 332. Barbora Gregusova is a Ph.D. student at Columbia University in music theory. She holds a B.M. in music theory and composition and a M.M. in music theory and musicology from the University of New Mexico. She

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served as an instructor of record for written music theory and musicianship courses in the undergraduate theory sequence at UNM. She has presented her work on transformational theory at several regional and graduate conference venues, including the Rocky Mountain Society for Music Theory, West Coast Conference for Music Theory and Analysis, Graduate Music Student Symposium (State University of New York at Buffalo), Composer’s Colloquium (UNM), and Music Theory Forum (Florida State University). She also presented her work at the Second International Festival of Czech Music hosted by the University of North Texas. Heather Holmquest is an instructor of music theory and voice at Umpqua Community College (Roseburg, Oregon). She received her Ph.D. in music theory at the University of Oregon, where she studied with Jack Boss and served as a Graduate Teaching Fellow in music theory and aural skills. She holds a M.A. in music theory from the University of Oregon, and a B.A. in music from Knox College. Her research interests include the interactions between analysis and performance practice, analysis of early music, reductive analysis, and ear training pedagogy. Recent projects have extended to analysis and socio-cultural explorations of popular and folk music, particularly in the works of Andrew Bird. She is also an active performer of early music, and constantly seeks to bring together her love of singing and music theory. Recent performances include collaborations with the Oregon Bach Collegium, the Jefferson Baroque Orchestra, and the Umpqua Community Orchestra, and she is a member of Vox Resonat, a premiere vocal ensemble that specializes in Medieval and Renaissance repertoire. Aaron J. Kirschner is a composer, theorist, clarinetist, and conductor, and adjunct professor of music theory at Utah Valley University. His music has been presented throughout the United States, as well as Italy and Finland, by members of the JACK Quartet, the Cortona Collective, and the American Modern Ensemble, among others. In July of 2014, he was Artist-in-Residence at the Arteles Creative Center in Haukijarvi, Finland. His theoretical research has been presented at the 2014 Society for Music Theory annual meeting, as well as at the Rocky Mountain regional SMT meeting and the West Coast Conference of Music Theory and Analysis. As a performer, Dr. Kirschner is a strong advocate for new music. Since making his solo debut in 2010 with the Boston New Music Initiative, Dr. Kirschner has premiered dozens of new works as a clarinetist and conductor. He holds a Ph.D. and M.M. in composition from the University

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of Utah and Boston University, respectively, as well as a B.M. in clarinet performance from the University of Iowa. Russell Knight earned a PhD in Music Theory from the University of California at Santa Barbara (2008), where he specialized in the music of Arnold Schoenberg. His dissertation, "Operand-Set Theory and Schoenberg's Erwartung, Op. 17" develops an analytical model for examining intervallic formations in compositions from Schoenberg's enigmatic middle period. Russ has presented papers at several conferences in the US and abroad, including regional meetings for the Society of Music Theory and the American Musicological Society, the National Meeting for the American Comparative Literature Association, the international meeting of the Society for Music Analysis and the International Conference on Music Since 1900. His research expertise includes post-tonal analytical methodology, manuscript studies, critical theory and computer applications in analysis. His recent article, "Common-Tone Tonality and Schubert's Ihr Bild: A Musical Parergon" appears in Ars Lyrica: Journal of the Lyrica Society for Word-Music Relations, vol 18 (2010). Russ has been on the faculty in music theory at San Diego State University, Northern Arizona University and the California Institute of the Arts. He is currently Lecturer in music theory and aural skills at University of California, Irvine. Rich Pellegrin holds the Ph.D. in music theory from the University of Washington. His dissertation, “On Jazz Analysis: Schenker, Salzer, and Salience” (2013), examines the significance of the Salzerian analytical tradition with respect to both the classical and jazz idioms. Dr. Pellegrin has presented research at numerous regional and national conferences, and is currently preparing work on Thelonious Monk for publication. He presently serves as Assistant Teaching Professor of Music Theory at the University of Missouri. Active as a jazz pianist and composer, Pellegrin’s debut record, ThreePart Odyssey, was released on Origin Records’ OA2 label in 2011. All About Jazz praised it as being “remarkably original” and a “superb debut,” giving it four and a half stars. With his band in Seattle (the Rich Pellegrin Quintet) he has recently completed a second album, Episodes IV-VI. Adam Shanley is currently in the fifth year of his doctoral studies at the University of Oregon in the music theory program, with a secondary area in intermedia music technology. He has presented papers at the West

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Contributors


Coast Conference of Music Theory and Analysis, as well as the Nief-Norf Research Summit and the Rocky Mountain Society for Music Theory. His research interests include experimental, atonal rock and electronic music as well as 12-tone, serial, and post-tonal composition. Adam has begun work on his dissertation, which focuses on Webern’s Op. 18, its use of the guitar, and its impact on the composer's serial works. Other recent writings have included a paper that explores structural cohesion across the recorded output of the psychedelic pop band “of Montreal," which he is currently preparing for journal submission. Inés Thiebaut, Visiting Assistant Professor at the University of Utah, is a PhD candidate in Composition (ABD) at the CUNY Graduate Center (New York), where she is currently writing her dissertation on the music of Mario Davidovsky with Joseph Straus as her advisor. She holds a Bachelor of Music degree in music theory from the Professional Conservatory of Music Adolfo Salazar (Madrid, Spain), a Composition and Film Scoring Bachelor of Music degree from Berklee College of Music (Boston) and a Master of Arts in Composition from Queens College (New York). While at CUNY she studied with composers Jason Eckardt, Douglas Geers, Jeff Nichols and Hubert Howe. Inés has also studied with composers Fabián Panisello and Marcela Rodriguez. Her music has been performed by multiple ensembles in the United States, and internationally in Spain, Portugal, and Mexico. She is also the co-founder and executive co-director of Dr. Faustus, an organization dedicated to the commission and promotion of new music. For more information on her music and her projects, you can visit her website: www.inesthiebaut.com. Dale Tovar is an 18-year-old graduate music theory student at the University of Oregon. He recently graduated Summa Cum Laude from Eastern Oregon University in 2015, where he completed a triple emphasis in music theory, jazz guitar, and classical guitar. He spent his junior year at the University of Utah through the National Student Exchange (NSE) program. Selected first from among over 2200 exchange students, he won the NSE’s 2015 Bette Worley (Outstanding) Achievement Award. His scholarly research relates to post-tonal voice leading, some of which is summarized in his 120-page undergraduate thesis “Hearing Voices: Reconciling Theories of Structure and Voice Leading and their Application to Atonal and Other Music.” In addition to this thesis, he has presented four distinct music theory research presentations, and a fifth is in progress.

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A performer of note, he has played advanced classical guitar repertoire, played in a graduate level jazz combo at the age of 16, and transcribed and played a virtuosic Michael Brecker solo on guitar. He has won several awards for his jazz guitar playing. He has studied classical guitar with Dr. Tully Cathey and Grand Ronde Symphony conductor and Vatican composer Dr. Leandro Espinosa. He has studied jazz with Geoffrey Miller, Bruce Forman, Dan Balmer, and Thelonious Monk Competition finalist Dr. Matt Cooper. He has studied music theory and done research under the guidance of Dr. Bruce Quaglia. He is a member of the Society of Music Theory, the Oregon Music Teachers Association, and the Phi Kappa Phi Honor Society. Yi-Cheng Daniel Wu completed his Ph.D. (2012) in Music Theory at the University at Buffalo, where he studied with Martha Hyde. His research interests focus on the topics of musical form, harmony, voice leading, and pitch contour in 20th- and 21st-century music. Prior to coming to Soochow University School of Music (Suzhou, China) in Fall 2013 as Assistant Professor of Music Theory and Curriculum Coordinator, he taught at Wesleyan University (Middletown CT, USA), where he served as the Visiting Assistant Professor of Music Theory. He has presented research at several theory conferences—such as Music Theory Society of New York State (Ithaca/USA, 2008; Binghamton/USA, 2015), Royal Music Association (Aberdeen/UK 2008; London/UK 2013), European Music Analysis (Italy, 2011; Belgium, 2014), Music Theory Midwest (Ann Arbor/USA, 2012), International Conference on Music Perception and Cognition (Seoul/South Korea, 2014), and West Coast Conference of Music Theory and Analysis (Salt Lake City/USA, 2014). His most recent paper on the topic of pitch contour will be coming out in the journal Indiana Theory Review. Aside from music theory, he is also interested in piano performance. In Spring of 2009, he received first prize in the 2008-2009 UBSO Concerto Competition, in which he performed the first movement from Saint-Saëns’s Second Piano Concerto. Brent Yorgason is an Associate Professor at Brigham Young University. He received his Ph.D. in music theory from Indiana University in 2009 with a dissertation entitled “Dispersal, Downbeat Space, and Metric Drift: Aspects of Expressive Timing and Meter.” He has presented his research in numerous regional, national, and international conferences, including the Society for Music Theory, Music Theory Midwest, the Rocky Mountain Society for Music Theory, the West Coast Conference of Music Theory and Analysis, the Association for Music in Technology, the IU

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Symposium on Performance and Analysis, the International Conference on Nineteenth-Century Music, and the symposium Performing Brahms in the Twenty-First Century in Leeds, England. His work has been published in Music Theory Pedagogy Online and in the collections Musical Currents from the Left Coast and Analyzing the Music of Living Composers. In addition to his studies of meter and performance, Brent’s research interests include such diverse topics as Schenkerian analysis, machine metaphors in music, minimalism and post-minimalism, the use of technology in music pedagogy, and hymnology. Brent is the Managing Editor of Music Theory Online, the official online journal of the Society for Music Theory—a position he has held since 2003. He is the creator and moderator of the society’s online discussion website, SMT Discuss, and serves on the SMT Networking Committee. He has also worked extensively as a computer programmer, developing music pedagogical software such as the Variations Audio Timeliner, ear training software, and other music-analytical tools. Brent has also helped to develop online music fundamentals courses for Indiana University (Music Fundamentals Online) and Connect4Education (OnMusicFundamentals).